Research ArticleOn Normal 119902-Ary Codes in Rosenbloom-Tsfasman Metric
R S Selvaraj and Venkatrajam Marka
Department of Mathematics National Institute of Technology Warangal Andhra Pradesh 506004 India
Correspondence should be addressed to R S Selvaraj rsselvanitwacin
Received 10 February 2014 Accepted 17 March 2014 Published 2 April 2014
Academic Editors A Cossidente E Manstavicius and S Richter
Copyright copy 2014 R S Selvaraj and V Marka This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The notion of normality of codes in Hamming metric is extended to the codes in Rosenbloom-Tsfasman metric (RT-metric inshort) Using concepts of partition number and 119897-cell of codes in RT-metric we establish results on covering radius and normalityof 119902-ary codes in this metric We also examine the acceptability of various coordinate positions of 119902-ary codes in this metric Andthus by exploring the feasibility of applying amalgamated direct summethod for construction of codes we analyze the significanceof normality in RT-metric
1 Introduction
Covering properties of codes have unique significance incoding theory and covering radius one of the four funda-mental parameters of a code is important in several respects[1] Considering the fact that it is a geometric property ofcodes that characterizes maximal error correcting capabilityin the case of minimum distance decoding covering radiushad been extensively studied by many researchers (see eg[2 3] and the literature therein) especially with respect tothe conventional Hamming metric In fact it has evolvedinto a subject in its own right mainly because of its practicalapplicability in areas such as data compression testing andwrite-once memories and also because of the mathematicalbeauty that it possesses More on covering radius can befound in the monograph compiled by Cohen et al [4]In order to improve upon the bounds on covering radiusvarious construction techniques that use two or more knowncodes to construct a new code were proposed over the lastfew decades One such method is direct sum constructionwhich is a basic yet useful construction method To improveupon the bounds related to covering radius of codes obtainedusing this method the notion of normality which facilitatesa construction technique known as amalgamated direct sum(ADS)was introduced for binary linear codes byGraham andSloane in [5] The same concepts were extended to binary
nonlinear codes by Cohen et al in [6] Later Lobstein andvan Wee generalized these results to 119902-ary codes [7]
In the present paper we extend the notion of normalityto codes in Rosenbloom-Tsfasman metric (RT-metric inshort) The present work is a generalization of our workon binary codes [8] to 119902-ary case (though the discussedresults hold good for codes over any alphabet of size 119902 forsimplicity we have taken it to be the finite field F119902) RT-metric was introduced by Rosenbloom and Tsfasman in [9]and independently by Skriganov in [10] and is more adequatethan Hamming metric in dealing with channels in whicherrors have a tendency to occur with a periodic spikewiseperturbation of period 119904 isin N As a generalization of theclassical Hammingmetric with richmathematical beauty andbeing advantageous over Hamming metric in dealing withcertain channels [9 10] this metric attracted the attentionof coding theorists over the last two decades [11ndash13] Thecovering problem in RT-metric was first dealt with by Yildizet al for RT-spaces over Galois Rings [14]
The organization of the present paper is as follows InSection 2 some basic definitions and notations that are usedin this paper are presented In Section 3 we have studied thecovering radius of 119902-ary codes in this metric by introducingtwo new tools namely partition number and 119897-cell whichgreatly reduce the difficulty in determining the coveringradius In Section 4 we investigate the normality of 119902-ary
Hindawi Publishing CorporationISRN CombinatoricsVolume 2014 Article ID 237915 5 pageshttpdxdoiorg1011552014237915
2 ISRN Combinatorics
codes in RT-metric In this section we also discuss thepossibilities of extending the direct sum and amalgamateddirect sum (ADS) constructions from Hamming metric toRT-metric And finally Section 5 provides the conclusions ofthis paper
2 Definitions and Notations
For 119909 = (1199091 1199092 119909119904) isin F 119904119902 where F119902 = GF(119902) is a finite field
of 119902 elements we define the 120588-weight (RT-weight) of 119909 to be
119908119905120588 (119909) = max 119894 | 119909119894 = 0 1 le 119894 le 119904 (1)
The RT-distance or 120588-distance between 119909 = (1199091 1199092 119909119904)
and 119910 = (1199101 1199102 119910119904) isin F 119904119902can be defined by 119889120588(119909 119910) =
119908119905120588(119909minus119910)The subsets of the space equippedwith thismetricare called RT-metric codes over F119902 (or 119902-ary RT-metric codes)and the subspaces are called linear RT-metric codes over F119902The value 119889min = min119889120588(119909 119910) 119909 119910 isin 119862 is called theminimum 120588-distance of the code 119862 The maximum distanceof any word in the ambient space from an RT-metric code iscalled the covering radius of the code A code in which eachnonzero codeword is of the same weight 119908120588 is said to be aconstant weight code A 119902-ary code of length 119904 cardinality 119870andminimum120588-distance119889120588 is said to be amaximumdistanceseparable code (an MDS code in short) if119870 = 119902
(119904minus119889120588+1)Throughout this paper unless otherwise specified by a
code we mean a 119902-ary RT-metric code and by distance wemean RT-distance Moreover [119904 119896 119889120588]119902119877 denotes a 119902-arylinear code of length 119904 dimension 119896 minimum distance 119889120588and covering radius 119877 (119904 119870 119889120588)119902119877 denotes a 119902-ary code withcardinality 119870 and 119905119902[119904 119896] denotes the minimum coveringradius that a 119902-ary linear code of length 119904 and dimension 119896
can possess
Definition 1 (direct sumof two codes) Let1198621 and1198622 be codeswith parameters (1199041 1198701)119902 and (1199042 1198702)119902 respectively Thenthe direct sum of 1198621 and 1198622 denoted by 1198621⨁1198622 is definedas
and is a code of length 1199041 + 1199042 and cardinality11987011198702
Definition 2 (119902-ary normal codes) Let119862be a 119902-aryRT-metriccode of length 119904 cardinality 119870 and covering radius 119877 andalso for each 119886 isin F119902 let 119862
(119894)
119886denote the set of codewords in
which the 119894th coordinate is 119886Then thenorm of119862with respectto the 119894th coordinate is defined as
119873(119894)
= max119909isinF 119904119902
sum
119886isinF119902
119889120588 (119909 119862(119894)
119886)
(3)
where
119889120588 (119909 119862(119894)
119886) =
min 119889120588 (119909 119910) | 119910 isin 119862(119894)
119886 if 119862(119894)
119886= 0
119904 if 119862(119894)119886
= 0(4)
Now 119873 = min119894119873(119894) is called the norm of 119862 and the
coordinates 119894 for which 119873(119894)
= 119873 are said to be acceptable
Finally a code is said to be normal if its norm satisfies 119873 le
119902119877 + 119902 minus 1 If the code 119862 is not clear from the context we usethe notations119873(119894)(119862) and119873(119862)
3 Covering Radius of RT-Metric Codes over F119902
Definition 3 (partition number of a 119902-ary RT-metric code)Let 119862 be an (119904 119870 119889120588)119902 code in RT-metric The largestnonnegative integer 119897 for which each 119902-ary 119897-tuple can beassigned to at least one codeword whose last 119897 coordinates areactually that 119897-tuple is called the partition number of the code119862 The code with partition number 119897 can be partitioned into119902119897 parts each of which has the property that all its membershave the same 119902-ary 119897-tuple as their last 119897 coordinates A partobtained in the above fashion is called an 119897-cell of the codeif at least one field element is not present in the (119904 minus 119897)thcoordinate of its member codewords As the definition ofpartition number suggests a code with partition number 119897
contains at least 119902119897 codewordsIf 119862 is an [119904 119896 119889120588]119902 linear code then each of the 119902
119897 partsdoes have 119902
119896minus119897 codewords in it and so does each 119897-cell by itsdefinition If1198621 is an 119897-cell of the linear code119862 then 119909+1198621 for119909 isin 1198621198621 will also be an 119897-cell different from1198621Thus for lin-ear codes if there is one 119897-cell then therewill be 119902119897 such 119897-cells
Now we will observe that covering radius of a code canbe determined using the concepts of partition number and119897-cell The definition of partition number serves as a tool indetermining the covering radius of an RT-metric code asshown by the following theorem
Theorem 4 Let 119862 be an (119904 119870 119889120588)119902119877 code in RT-metric Thenthe partition number of 119862 is 119897 if and only if its covering radiusis 119904 minus 119897
Proof First let the partition number of the code be 119897Partition the code such that codewords in each part containa unique 119902-ary (119897 + 1)-tuple as their last 119897 + 1 coordinatesAssociate each part with the respective (119897+1)-tuple Since thepartition number of the code is 119897 the number of such partswill be less than 119902
119897+1 Choose an 119909 isin F 119904119902 whose last 119897 + 1
coordinates constitute the (119897 + 1)-tuple that does not have apart associated with it Such an 119909 will be at distance 119904 minus 119897
from the code which is the maximum distance that a wordcan actually be from the code 119862 Hence the covering radiusis 119904 minus 119897
Conversely let the covering radius be 119877 = 119904 minus 119897 By thedefinition of covering radius any word must be at distance atmost 119904 minus 119897 from the code which means that for each wordthere is at least one codeword which agrees with the word inthe last 119904 minus (119904 minus 119897) = 119897 coordinate positions This implies thatthe partition number of the code is greater than or equal to 119897Let us assume that the partition number of the code is 119898 gt
119897 Then to each 119902-ary 119898-tuple we can associate a codewordwhose last 119898 coordinates coincide with that 119898-tuple Thuseachword in F 119904
119902will be at distance atmost 119904minus119898 contradicting
the fact that covering radius of 119862 is 119904 minus 119897 Hence the partitionnumber of the code 119862 is 119897
ISRN Combinatorics 3
Now the above theorem can be restated in terms of 119897-cell in the following manner as the definition of the 119897-cellsuggests
Corollary 5 Let 119862 be an RT-metric code of length 119904 over F119902Then the covering radius of 119862 is 119904 minus 119897 if and only if exist an 119897-cellof the code 119862
Proposition 6 Let 1198621 and 1198622 be any two RT-metric codesover F119902 with parameters (1199041 1198701 1198891)1199021198771 and (1199042 1198702 1198892)1199021198772respectively Then their direct sum 119862 = 1198621⨁1198622 is an (1199041 +
Clearly this is a code of length 1199041 + 1199042 and cardinality 11987011198702Now since the minimum distance of 1198621 is 1198891 there exist twocodewords 1198881 and 119888
1015840
1in 1198621 such that 119889120588(1198881 119888
1015840
1) = 1198891 Then
the codewords 119888 = (1198881 1198882) and 1198881015840= (1198881015840
1 1198882) for some 1198882 isin 1198622
will also be at distance 1198891 which is minimum among all thecodewords of 119862 FromTheorem 4 and Definition 3 it is clearthat the covering radius of a code in RT-metric depends onthe partition number of the code and that the process ofpartitioning starts with the right most coordinate ThereforebyTheorem 4 unless1198622 is the space F
1199042119902 the partition number
of 119862 must be equal to that of 1198622 which is 1199042 minus 1198772 and hencethe covering radius of 119862 is 1199041 + 1199042 minus (1199042 minus 1198772) = 1199041 + 1198772
Remark 7 If 1198622 = F 1199042119902 then the covering radius of the code
119862 = 1198621⨁1198622 is 119877 = 1198771
4 Normality of Codes in RT-Metric
Proposition 8 If 119862 is an (119904 119870 119889120588)119902119877 RT-metric code over F119902then119873(119862) ge 119902119877
Proof The proof follows directly from the definition ofcovering radius and that of norm of a code
Theorem9 Any 119902-ary RT-metric code of length 119904 and coveringradius 119904 is normal and all the coordinates are acceptable
Proof As 119877 = 119904 the partition number is 0 which meansthat there exists an 120572 isin F119902 which is not present as the lastcoordinate of any codeword in 119862 If 119909 = (1199091 1199092 119909119904) isin F 119904
119902
is such that 119909119904 = 120572 then 119889120588(119909 119862(119894)
119886) = 119904 for each 119886 isin F119902
and for any 119894 isin 1 2 119904 Thus 119873(119894) = 119902119904 lt 119902119904 + 119902 minus 1 =
119902119877+(119902minus1) Hence the code is normal and all the coordinatesare acceptable
Lemma 10 Let 119862 be any RT-metric code with length 119904 andcovering radius 119877 lt 119904 then
119873(119894)
= (119902 minus 1) 119894 + 119877 (6)
for each 119894 isin 119877 + 1 119877 + 2 119904
Proof For a coordinate position 119894 isin 119877 + 1 119877 + 2 119904 thenorm is given by
119873(119894)
= max119909isinF 119904119902
sum
119886isinF119902
119889120588 (119909 119862(119894)
119886)
(7)
As the covering radius of 119862 is 119877 it has partition number119904 minus 119877 So the definition of partition number implies thateach codeword can be associated with a unique (119904 minus 119877)-tuplewhich actually consists of the last (119904 minus 119877) coordinates of thatcodeword and also that there exists at least one (119904 minus 119877 + 1)-tuple which is not the same as the last 119904 minus 119877 + 1 coordinatesof any of the codewords Now when we partition the codeinto 119902 parts 119862(119894)
119886 for 119886 isin F119902 we can also partition the set of
all (119904 minus 119877)-tuples into 119902 parts corresponding to the associatedcodewords in 119862
(119894)
119886 If we choose a word 119909 isin F 119904
119902whose last
119904 minus 119877 + 1 coordinates do not match with the last 119904 minus 119877 + 1
coordinates of any of the codewords then for this wordsum119886isinF119902
119889120588(119909 119862(119894)
119886) = (119902 minus 1)119894 + 119877 which is the maximum value
that a word can give Hence the proof holds
Theorem 11 Let 119862 be any RT-metric code of length 119904 withcovering radius 119877 lt 119904 Then the coordinates 119894 isin 119877 + 2 119877 +
3 119904 are not acceptable
Proof Theminimum norm is
119873 le 119873(119894) forall119894 isin 1 2 119904 (8)
By Lemma 10 119873(119894) = (119902 minus 1)119894 + 119877 for all 119894 ge 119877 + 1 whichimplies
The above theorem is not sufficient to arrive at a decisionon the acceptability of the 119904th coordinate when the codehas covering radius 119877 = 119904 minus 1 This can be settled by thefollowing theorem which says that no 119902-ary linear RT-metriccode with dimension more than 1 and covering radius 119904 minus 1
has acceptable last coordinate
Theorem 12 Let 119862 be an [119904 119896 119889120588]119902119877 linear RT-metric codeover F119902 with 119896 gt 1 and 119877 = 119904 minus 1 Then the last coordinate isnot acceptable
Proof By Lemma 10
119873(119904)
= (119902 minus 1) 119904 + 119877
= (119902 minus 1) 119904 + 119904 minus 1 = 119902119904 minus 1
= 119902 (119904 minus 1) + 119902 minus 1 = 119902119877 + (119902 minus 1)
(since 119877 = 119904 minus 1)
(11)
4 ISRN Combinatorics
In order to prove this theorem we must show that thereexists at least one coordinate 119895 for which 119873
(119895)lt 119873(119904) As
the dimension of 119862 is 119896 there will be 119896 coordinate positions1198941 1198942 119894119896 such that 119862
(119894119905)
119886= 0 for all 119886 isin F119902 and for each
119905 = 1 2 119896 One such coordinate position is 119904 (but not 119904minus1as the partition number is 1) As 119896 gt 1 there exists at leastonemore such coordinate position 119895 isin 1198941 1198942 119894119896 and 119895 = 119904
wherein 119862(119895)
119886= 0 for all 119886 isin F119902 and so 119873
(119895)le 119902(119904 minus 1) lt 119873
(119904)This completes the proof
But when the code is either of dimension 1 or of constantweight all the coordinates become acceptable as the followingresults suggest
Proposition 13 Let 119862 be an [119904 1]119902 linear RT-metric codeThen 119862 is normal and all the coordinates are acceptable
Proof It can easily be seen that the covering radius of 119862 iseither 119904 or 119904minus1 If it is 119904 the result follows byTheorem 9 If it is119904minus1 there exists a 1-cell Since119862 is of dimension 1 there exist119902 such 1-cells By Lemma 10 119873(119904) = 119902119904 minus 1 Now for each 119894 isin
1 2 119904minus1 either119862(119894)0
= 119862 or |119862(119894)0| = 1 In the former case
119862(119894)
119886= 0 for all 119886 = 0 and in the latter case |119862(119894)
119886| = 1 for all 119886 isin
F119902 In any case119873(119894)
= 119902119904minus1This proves the result stated
Proposition 14 Let 119862 be a constant weight code of length119904 over F119902 Then 119862 is normal and all the coordinates areacceptable
Proof It is easy to see that the covering radius 119877 of theconstant weight code 119862 is either 119904 or 119904 minus 1 as it can havepartition number of either 0 or 1 If119877 = 119904 then by Lemma 10it is done If 119877 = 119904 minus 1 then 119862
(119904)
119886= 0 for any 119886 isin F119902 Then for
any 119894 0 isin 119862(119894)
0and |119862
(119894)
0| ge 1 For an 119909 isin F 119904
119902with 119909119904 = 0 and
119909119904minus1 = 0119873(119894) equals 119902119904 minus 1 Hence the result follows
Remark 15 If 119862 = F 119904119902 then its partition number is 119904 and so its
covering radius 119877 is 0 By (3) for any 119894 119873(119894) = (119902 minus 1)119894 + 0 =
(119902 minus 1)119894 And so119873(1) lt 119873(2)
lt sdot sdot sdot lt 119873(119904) and119873
(1)= 119902 minus 1 =
119902119877+119902minus1 Thus the ambient space F 119904119902is normal with the first
coordinate being the only acceptable coordinate
41 Normality and Direct Sum Construction
Proposition 16 Let 1198621 and 1198622 be (1199041 1198701 1198891)1199021198771 and(1199042 1198702 1198892)1199021198772 RT-metric codes respectively such that 1198772 = 0Then
(i) the direct sum 119862 = 1198621⨁1198622 is normal(ii) all the coordinates corresponding to 1198621 are acceptable
for 119862(iii) the coordinate 119894 for 1199041 + 1 le 119894 le 1199042 is accept-
able for 119862 only if the norm of 1198622 with respect to thecoordinate 119894 minus 1199041 is equal to 1199021198772
Proof Let 1198621 and 1198622 be any (1199041 1198701 1198891)1199021198771 and (1199042
1198702 1198892)1199021198772 119902-ary RT-metric codes respectively Then by
Proposition 6 their direct sum 119862 = 1198621⨁1198622 is an (119904 11987011198702
1198891)119902119877 code with length 119904 = 1199041 + 1199042 and covering radius119877 = 1199041 + 1198772 Now for 119894 isin 1 2 1199041 by (3) we have
119873(119894)
(119862) = 119902 (1199041 + 1198772) (12)
Now in order to determine the norm 119873(119862) we have to find119873(119895)(119862) for each 119895 isin 1199041 +1 1199041 +2 1199041 + 1199042 the coordinate
positions corresponding to1198622 But we know by Proposition 8that119873(1198622) ge 1199021198772 which implies119873(119895)(119862) ge 119902(1199041+1198772) Hence119873(119862) = 119902(1199041+1198772) = 119902119877 lt 119902119877+119902minus1Thus119862 is normal and allthe coordinate positions corresponding to 1198621 are acceptableand the coordinate position 119895 isin 1199041 + 1 1199041 + 2 1199041 + 1199042
corresponding to 1198622 is acceptable if119873(119895minus1199041)(1198622) = 1199021198772
42 Normality and Amalgamated Direct Sum ConstructionThe amalgamated direct sum of two codes is defined asfollows
Definition 17 Let 1198621 be an [1199041 1198961]1199021198771 normal code with thelast coordinate being acceptable and let 1198622 be an [1199042 1198962]1199021198772normal code with the first coordinate being acceptableThentheir amalgamated direct sum (or shortly ADS) denoted by1198621⨁1198622 is an [1199041 + 1199042 minus 1 1198961 + 1198962 minus 1]119902(1199041 + 1198772 minus 1) code andis given by
Here in this definition we consider those codes 1198621 and 1198622which are normal respectively with the last coordinate andfirst coordinate being acceptable such that the parts 119862(119904)
1119886and
119862(1)
2119886are nonempty for all 119886 isin F119902
As Theorem 12 shows the nonexistence of linear codes ofdimension more than 1 whose last coordinate is acceptablethe significance of amalgamated direct sum and hence thatof normality in RT-metric are less as far as the resultingcovering radius is concerned But since the one-dimensionalcodes are normal and all the coordinates are acceptable onecan only use this ADS construction method to combine a1-dimensional code with any other linear code which maybe helpful in the construction of MDS codes as seen in thefollowing result
Theorem 18 Let 1198621 be an [1199041 1]119902 MDS code with coveringradius 119905119902[1199041 1] (which is normal with last coordinate beingacceptable) and let 1198622 be an [1199042 1198962]119902 MDS code with coveringradius 119905119902[1199042 1198962] Then their amalgamated direct sum 1198621⨁1198622is an [1199041 + 1199042 minus 1 1198962]119902 MDS code with covering radius 119905119902[1199041 +1199042 minus 1 1198962]
Proof From the definition of partition number and fromTheorem 4 it is easy to see that 119905119902[119904 119896] = 119904 minus 119896 So 119905119902[1199041 1] =1199041 minus 1 and 119905119902[1199042 1198962] = 1199042 minus 1198962 Now as the dimension of1198621⨁1198622 is 1198962 (which is less than or equal to 1199042) the coveringradius of1198621⨁1198622 depends only on the coordinates pertainingto the code 1198622 But 1198622 has covering radius 1199042 minus 1198962 and hence
ISRN Combinatorics 5
has partition number 1198962 So 1198621⨁1198622 also will have partitionnumber 1198962 and hence will have covering radius 1199041+1199042minus1minus1198962
which is equal to 119905119902[1199041+1199042minus1 1198962] which implies that1198621⨁1198622is MDS This completes the proof
5 Conclusion
We discussed the normality of codes over F119902 in RT-metricand determined its norm with respect to various coordinatepositions We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Delsarte ldquoFour fundamental parameters of a code and theircombinatorial significancerdquo Information and Control vol 23no 5 pp 407ndash438 1973
[2] G D Cohen M G Karpovsky H F Mattson and J R SchatzldquoCovering radiusmdashsurvey and recent resultsrdquo IEEE Transac-tions on Information Theory vol 31 no 3 pp 328ndash343 1985
[3] G D Cohen S N Litsyn A C Lobstein and H F Mattson JrldquoCovering radius 1985ndash1994rdquoApplicable Algebra in EngineeringCommunications and Computing vol 8 no 3 pp 173ndash239 1997
[4] G D Cohen I Honkala S Litsyn and A Lobstein CoveringCodes Elsevier North-Holland The Netherlands 1997
[5] R L Graham and N J A Sloane ldquoOn the covering radius ofcodesrdquo IEEE Transactions on Information Theory vol 31 no 3pp 385ndash401 1985
[6] GD Cohen A C Lobstein andN J A Sloane ldquoFurther resultson the covering radius of codesrdquo IEEE Transactions on Informa-tion Theory vol 32 no 6 pp 680ndash694 1986
[7] AC Lobstein andG JM vanWee ldquoOnnormal and subnormalq-ary codesrdquo IEEE Transactions on Information Theory vol 35no 6 pp 1291ndash1295 1989
[8] R S Selvaraj and V Marka ldquoNormality of binary codes inRosenbloom-Tsfasman metricrdquo in Proceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo13) pp 1370ndash1373 August 2013
[9] M Yu Rosenbloom andMA Tsfasman ldquoCodes for them-met-ricrdquo Problems of Information Transmission vol 33 pp 45ndash521997
[10] M M Skriganov ldquoCoding theory and uniform distributionsrdquoSt Petersburg Mathematical Journal vol 143 no 2 pp 301ndash3372002
[11] I Siap and M Ozen ldquoThe complete weight enumerator forcodes over 119872
119899times119904 (119877) with respect to the Rosenbloom-Tsfasmanmetricrdquo Applied Mathematics Letters vol 17 no 1 pp 65ndash692004
[12] S TDougherty andK Shiromoto ldquoMaximumdistance codes in119872119886119905119899times119904
(Z119896)with a non-hammingmetric and uniformdistribu-
tionsrdquo Designs Codes and Cryptography vol 33 no 1 pp 45ndash61 2004
[13] S Jain ldquoCT burstsmdashfrom classical to array codingrdquo DiscreteMathematics vol 308 no 9 pp 1489ndash1499 2008
[14] B Yildiz I Siap T Bilgin and G Yesilot ldquoThe covering prob-lem for finite rings with respect to the RT-metricrdquo AppliedMathematics Letters vol 23 no 9 pp 988ndash992 2010
codes in RT-metric In this section we also discuss thepossibilities of extending the direct sum and amalgamateddirect sum (ADS) constructions from Hamming metric toRT-metric And finally Section 5 provides the conclusions ofthis paper
2 Definitions and Notations
For 119909 = (1199091 1199092 119909119904) isin F 119904119902 where F119902 = GF(119902) is a finite field
of 119902 elements we define the 120588-weight (RT-weight) of 119909 to be
119908119905120588 (119909) = max 119894 | 119909119894 = 0 1 le 119894 le 119904 (1)
The RT-distance or 120588-distance between 119909 = (1199091 1199092 119909119904)
and 119910 = (1199101 1199102 119910119904) isin F 119904119902can be defined by 119889120588(119909 119910) =
119908119905120588(119909minus119910)The subsets of the space equippedwith thismetricare called RT-metric codes over F119902 (or 119902-ary RT-metric codes)and the subspaces are called linear RT-metric codes over F119902The value 119889min = min119889120588(119909 119910) 119909 119910 isin 119862 is called theminimum 120588-distance of the code 119862 The maximum distanceof any word in the ambient space from an RT-metric code iscalled the covering radius of the code A code in which eachnonzero codeword is of the same weight 119908120588 is said to be aconstant weight code A 119902-ary code of length 119904 cardinality 119870andminimum120588-distance119889120588 is said to be amaximumdistanceseparable code (an MDS code in short) if119870 = 119902
(119904minus119889120588+1)Throughout this paper unless otherwise specified by a
code we mean a 119902-ary RT-metric code and by distance wemean RT-distance Moreover [119904 119896 119889120588]119902119877 denotes a 119902-arylinear code of length 119904 dimension 119896 minimum distance 119889120588and covering radius 119877 (119904 119870 119889120588)119902119877 denotes a 119902-ary code withcardinality 119870 and 119905119902[119904 119896] denotes the minimum coveringradius that a 119902-ary linear code of length 119904 and dimension 119896
can possess
Definition 1 (direct sumof two codes) Let1198621 and1198622 be codeswith parameters (1199041 1198701)119902 and (1199042 1198702)119902 respectively Thenthe direct sum of 1198621 and 1198622 denoted by 1198621⨁1198622 is definedas
and is a code of length 1199041 + 1199042 and cardinality11987011198702
Definition 2 (119902-ary normal codes) Let119862be a 119902-aryRT-metriccode of length 119904 cardinality 119870 and covering radius 119877 andalso for each 119886 isin F119902 let 119862
(119894)
119886denote the set of codewords in
which the 119894th coordinate is 119886Then thenorm of119862with respectto the 119894th coordinate is defined as
119873(119894)
= max119909isinF 119904119902
sum
119886isinF119902
119889120588 (119909 119862(119894)
119886)
(3)
where
119889120588 (119909 119862(119894)
119886) =
min 119889120588 (119909 119910) | 119910 isin 119862(119894)
119886 if 119862(119894)
119886= 0
119904 if 119862(119894)119886
= 0(4)
Now 119873 = min119894119873(119894) is called the norm of 119862 and the
coordinates 119894 for which 119873(119894)
= 119873 are said to be acceptable
Finally a code is said to be normal if its norm satisfies 119873 le
119902119877 + 119902 minus 1 If the code 119862 is not clear from the context we usethe notations119873(119894)(119862) and119873(119862)
3 Covering Radius of RT-Metric Codes over F119902
Definition 3 (partition number of a 119902-ary RT-metric code)Let 119862 be an (119904 119870 119889120588)119902 code in RT-metric The largestnonnegative integer 119897 for which each 119902-ary 119897-tuple can beassigned to at least one codeword whose last 119897 coordinates areactually that 119897-tuple is called the partition number of the code119862 The code with partition number 119897 can be partitioned into119902119897 parts each of which has the property that all its membershave the same 119902-ary 119897-tuple as their last 119897 coordinates A partobtained in the above fashion is called an 119897-cell of the codeif at least one field element is not present in the (119904 minus 119897)thcoordinate of its member codewords As the definition ofpartition number suggests a code with partition number 119897
contains at least 119902119897 codewordsIf 119862 is an [119904 119896 119889120588]119902 linear code then each of the 119902
119897 partsdoes have 119902
119896minus119897 codewords in it and so does each 119897-cell by itsdefinition If1198621 is an 119897-cell of the linear code119862 then 119909+1198621 for119909 isin 1198621198621 will also be an 119897-cell different from1198621Thus for lin-ear codes if there is one 119897-cell then therewill be 119902119897 such 119897-cells
Now we will observe that covering radius of a code canbe determined using the concepts of partition number and119897-cell The definition of partition number serves as a tool indetermining the covering radius of an RT-metric code asshown by the following theorem
Theorem 4 Let 119862 be an (119904 119870 119889120588)119902119877 code in RT-metric Thenthe partition number of 119862 is 119897 if and only if its covering radiusis 119904 minus 119897
Proof First let the partition number of the code be 119897Partition the code such that codewords in each part containa unique 119902-ary (119897 + 1)-tuple as their last 119897 + 1 coordinatesAssociate each part with the respective (119897+1)-tuple Since thepartition number of the code is 119897 the number of such partswill be less than 119902
119897+1 Choose an 119909 isin F 119904119902 whose last 119897 + 1
coordinates constitute the (119897 + 1)-tuple that does not have apart associated with it Such an 119909 will be at distance 119904 minus 119897
from the code which is the maximum distance that a wordcan actually be from the code 119862 Hence the covering radiusis 119904 minus 119897
Conversely let the covering radius be 119877 = 119904 minus 119897 By thedefinition of covering radius any word must be at distance atmost 119904 minus 119897 from the code which means that for each wordthere is at least one codeword which agrees with the word inthe last 119904 minus (119904 minus 119897) = 119897 coordinate positions This implies thatthe partition number of the code is greater than or equal to 119897Let us assume that the partition number of the code is 119898 gt
119897 Then to each 119902-ary 119898-tuple we can associate a codewordwhose last 119898 coordinates coincide with that 119898-tuple Thuseachword in F 119904
119902will be at distance atmost 119904minus119898 contradicting
the fact that covering radius of 119862 is 119904 minus 119897 Hence the partitionnumber of the code 119862 is 119897
ISRN Combinatorics 3
Now the above theorem can be restated in terms of 119897-cell in the following manner as the definition of the 119897-cellsuggests
Corollary 5 Let 119862 be an RT-metric code of length 119904 over F119902Then the covering radius of 119862 is 119904 minus 119897 if and only if exist an 119897-cellof the code 119862
Proposition 6 Let 1198621 and 1198622 be any two RT-metric codesover F119902 with parameters (1199041 1198701 1198891)1199021198771 and (1199042 1198702 1198892)1199021198772respectively Then their direct sum 119862 = 1198621⨁1198622 is an (1199041 +
Clearly this is a code of length 1199041 + 1199042 and cardinality 11987011198702Now since the minimum distance of 1198621 is 1198891 there exist twocodewords 1198881 and 119888
1015840
1in 1198621 such that 119889120588(1198881 119888
1015840
1) = 1198891 Then
the codewords 119888 = (1198881 1198882) and 1198881015840= (1198881015840
1 1198882) for some 1198882 isin 1198622
will also be at distance 1198891 which is minimum among all thecodewords of 119862 FromTheorem 4 and Definition 3 it is clearthat the covering radius of a code in RT-metric depends onthe partition number of the code and that the process ofpartitioning starts with the right most coordinate ThereforebyTheorem 4 unless1198622 is the space F
1199042119902 the partition number
of 119862 must be equal to that of 1198622 which is 1199042 minus 1198772 and hencethe covering radius of 119862 is 1199041 + 1199042 minus (1199042 minus 1198772) = 1199041 + 1198772
Remark 7 If 1198622 = F 1199042119902 then the covering radius of the code
119862 = 1198621⨁1198622 is 119877 = 1198771
4 Normality of Codes in RT-Metric
Proposition 8 If 119862 is an (119904 119870 119889120588)119902119877 RT-metric code over F119902then119873(119862) ge 119902119877
Proof The proof follows directly from the definition ofcovering radius and that of norm of a code
Theorem9 Any 119902-ary RT-metric code of length 119904 and coveringradius 119904 is normal and all the coordinates are acceptable
Proof As 119877 = 119904 the partition number is 0 which meansthat there exists an 120572 isin F119902 which is not present as the lastcoordinate of any codeword in 119862 If 119909 = (1199091 1199092 119909119904) isin F 119904
119902
is such that 119909119904 = 120572 then 119889120588(119909 119862(119894)
119886) = 119904 for each 119886 isin F119902
and for any 119894 isin 1 2 119904 Thus 119873(119894) = 119902119904 lt 119902119904 + 119902 minus 1 =
119902119877+(119902minus1) Hence the code is normal and all the coordinatesare acceptable
Lemma 10 Let 119862 be any RT-metric code with length 119904 andcovering radius 119877 lt 119904 then
119873(119894)
= (119902 minus 1) 119894 + 119877 (6)
for each 119894 isin 119877 + 1 119877 + 2 119904
Proof For a coordinate position 119894 isin 119877 + 1 119877 + 2 119904 thenorm is given by
119873(119894)
= max119909isinF 119904119902
sum
119886isinF119902
119889120588 (119909 119862(119894)
119886)
(7)
As the covering radius of 119862 is 119877 it has partition number119904 minus 119877 So the definition of partition number implies thateach codeword can be associated with a unique (119904 minus 119877)-tuplewhich actually consists of the last (119904 minus 119877) coordinates of thatcodeword and also that there exists at least one (119904 minus 119877 + 1)-tuple which is not the same as the last 119904 minus 119877 + 1 coordinatesof any of the codewords Now when we partition the codeinto 119902 parts 119862(119894)
119886 for 119886 isin F119902 we can also partition the set of
all (119904 minus 119877)-tuples into 119902 parts corresponding to the associatedcodewords in 119862
(119894)
119886 If we choose a word 119909 isin F 119904
119902whose last
119904 minus 119877 + 1 coordinates do not match with the last 119904 minus 119877 + 1
coordinates of any of the codewords then for this wordsum119886isinF119902
119889120588(119909 119862(119894)
119886) = (119902 minus 1)119894 + 119877 which is the maximum value
that a word can give Hence the proof holds
Theorem 11 Let 119862 be any RT-metric code of length 119904 withcovering radius 119877 lt 119904 Then the coordinates 119894 isin 119877 + 2 119877 +
3 119904 are not acceptable
Proof Theminimum norm is
119873 le 119873(119894) forall119894 isin 1 2 119904 (8)
By Lemma 10 119873(119894) = (119902 minus 1)119894 + 119877 for all 119894 ge 119877 + 1 whichimplies
The above theorem is not sufficient to arrive at a decisionon the acceptability of the 119904th coordinate when the codehas covering radius 119877 = 119904 minus 1 This can be settled by thefollowing theorem which says that no 119902-ary linear RT-metriccode with dimension more than 1 and covering radius 119904 minus 1
has acceptable last coordinate
Theorem 12 Let 119862 be an [119904 119896 119889120588]119902119877 linear RT-metric codeover F119902 with 119896 gt 1 and 119877 = 119904 minus 1 Then the last coordinate isnot acceptable
Proof By Lemma 10
119873(119904)
= (119902 minus 1) 119904 + 119877
= (119902 minus 1) 119904 + 119904 minus 1 = 119902119904 minus 1
= 119902 (119904 minus 1) + 119902 minus 1 = 119902119877 + (119902 minus 1)
(since 119877 = 119904 minus 1)
(11)
4 ISRN Combinatorics
In order to prove this theorem we must show that thereexists at least one coordinate 119895 for which 119873
(119895)lt 119873(119904) As
the dimension of 119862 is 119896 there will be 119896 coordinate positions1198941 1198942 119894119896 such that 119862
(119894119905)
119886= 0 for all 119886 isin F119902 and for each
119905 = 1 2 119896 One such coordinate position is 119904 (but not 119904minus1as the partition number is 1) As 119896 gt 1 there exists at leastonemore such coordinate position 119895 isin 1198941 1198942 119894119896 and 119895 = 119904
wherein 119862(119895)
119886= 0 for all 119886 isin F119902 and so 119873
(119895)le 119902(119904 minus 1) lt 119873
(119904)This completes the proof
But when the code is either of dimension 1 or of constantweight all the coordinates become acceptable as the followingresults suggest
Proposition 13 Let 119862 be an [119904 1]119902 linear RT-metric codeThen 119862 is normal and all the coordinates are acceptable
Proof It can easily be seen that the covering radius of 119862 iseither 119904 or 119904minus1 If it is 119904 the result follows byTheorem 9 If it is119904minus1 there exists a 1-cell Since119862 is of dimension 1 there exist119902 such 1-cells By Lemma 10 119873(119904) = 119902119904 minus 1 Now for each 119894 isin
1 2 119904minus1 either119862(119894)0
= 119862 or |119862(119894)0| = 1 In the former case
119862(119894)
119886= 0 for all 119886 = 0 and in the latter case |119862(119894)
119886| = 1 for all 119886 isin
F119902 In any case119873(119894)
= 119902119904minus1This proves the result stated
Proposition 14 Let 119862 be a constant weight code of length119904 over F119902 Then 119862 is normal and all the coordinates areacceptable
Proof It is easy to see that the covering radius 119877 of theconstant weight code 119862 is either 119904 or 119904 minus 1 as it can havepartition number of either 0 or 1 If119877 = 119904 then by Lemma 10it is done If 119877 = 119904 minus 1 then 119862
(119904)
119886= 0 for any 119886 isin F119902 Then for
any 119894 0 isin 119862(119894)
0and |119862
(119894)
0| ge 1 For an 119909 isin F 119904
119902with 119909119904 = 0 and
119909119904minus1 = 0119873(119894) equals 119902119904 minus 1 Hence the result follows
Remark 15 If 119862 = F 119904119902 then its partition number is 119904 and so its
covering radius 119877 is 0 By (3) for any 119894 119873(119894) = (119902 minus 1)119894 + 0 =
(119902 minus 1)119894 And so119873(1) lt 119873(2)
lt sdot sdot sdot lt 119873(119904) and119873
(1)= 119902 minus 1 =
119902119877+119902minus1 Thus the ambient space F 119904119902is normal with the first
coordinate being the only acceptable coordinate
41 Normality and Direct Sum Construction
Proposition 16 Let 1198621 and 1198622 be (1199041 1198701 1198891)1199021198771 and(1199042 1198702 1198892)1199021198772 RT-metric codes respectively such that 1198772 = 0Then
(i) the direct sum 119862 = 1198621⨁1198622 is normal(ii) all the coordinates corresponding to 1198621 are acceptable
for 119862(iii) the coordinate 119894 for 1199041 + 1 le 119894 le 1199042 is accept-
able for 119862 only if the norm of 1198622 with respect to thecoordinate 119894 minus 1199041 is equal to 1199021198772
Proof Let 1198621 and 1198622 be any (1199041 1198701 1198891)1199021198771 and (1199042
1198702 1198892)1199021198772 119902-ary RT-metric codes respectively Then by
Proposition 6 their direct sum 119862 = 1198621⨁1198622 is an (119904 11987011198702
1198891)119902119877 code with length 119904 = 1199041 + 1199042 and covering radius119877 = 1199041 + 1198772 Now for 119894 isin 1 2 1199041 by (3) we have
119873(119894)
(119862) = 119902 (1199041 + 1198772) (12)
Now in order to determine the norm 119873(119862) we have to find119873(119895)(119862) for each 119895 isin 1199041 +1 1199041 +2 1199041 + 1199042 the coordinate
positions corresponding to1198622 But we know by Proposition 8that119873(1198622) ge 1199021198772 which implies119873(119895)(119862) ge 119902(1199041+1198772) Hence119873(119862) = 119902(1199041+1198772) = 119902119877 lt 119902119877+119902minus1Thus119862 is normal and allthe coordinate positions corresponding to 1198621 are acceptableand the coordinate position 119895 isin 1199041 + 1 1199041 + 2 1199041 + 1199042
corresponding to 1198622 is acceptable if119873(119895minus1199041)(1198622) = 1199021198772
42 Normality and Amalgamated Direct Sum ConstructionThe amalgamated direct sum of two codes is defined asfollows
Definition 17 Let 1198621 be an [1199041 1198961]1199021198771 normal code with thelast coordinate being acceptable and let 1198622 be an [1199042 1198962]1199021198772normal code with the first coordinate being acceptableThentheir amalgamated direct sum (or shortly ADS) denoted by1198621⨁1198622 is an [1199041 + 1199042 minus 1 1198961 + 1198962 minus 1]119902(1199041 + 1198772 minus 1) code andis given by
Here in this definition we consider those codes 1198621 and 1198622which are normal respectively with the last coordinate andfirst coordinate being acceptable such that the parts 119862(119904)
1119886and
119862(1)
2119886are nonempty for all 119886 isin F119902
As Theorem 12 shows the nonexistence of linear codes ofdimension more than 1 whose last coordinate is acceptablethe significance of amalgamated direct sum and hence thatof normality in RT-metric are less as far as the resultingcovering radius is concerned But since the one-dimensionalcodes are normal and all the coordinates are acceptable onecan only use this ADS construction method to combine a1-dimensional code with any other linear code which maybe helpful in the construction of MDS codes as seen in thefollowing result
Theorem 18 Let 1198621 be an [1199041 1]119902 MDS code with coveringradius 119905119902[1199041 1] (which is normal with last coordinate beingacceptable) and let 1198622 be an [1199042 1198962]119902 MDS code with coveringradius 119905119902[1199042 1198962] Then their amalgamated direct sum 1198621⨁1198622is an [1199041 + 1199042 minus 1 1198962]119902 MDS code with covering radius 119905119902[1199041 +1199042 minus 1 1198962]
Proof From the definition of partition number and fromTheorem 4 it is easy to see that 119905119902[119904 119896] = 119904 minus 119896 So 119905119902[1199041 1] =1199041 minus 1 and 119905119902[1199042 1198962] = 1199042 minus 1198962 Now as the dimension of1198621⨁1198622 is 1198962 (which is less than or equal to 1199042) the coveringradius of1198621⨁1198622 depends only on the coordinates pertainingto the code 1198622 But 1198622 has covering radius 1199042 minus 1198962 and hence
ISRN Combinatorics 5
has partition number 1198962 So 1198621⨁1198622 also will have partitionnumber 1198962 and hence will have covering radius 1199041+1199042minus1minus1198962
which is equal to 119905119902[1199041+1199042minus1 1198962] which implies that1198621⨁1198622is MDS This completes the proof
5 Conclusion
We discussed the normality of codes over F119902 in RT-metricand determined its norm with respect to various coordinatepositions We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Delsarte ldquoFour fundamental parameters of a code and theircombinatorial significancerdquo Information and Control vol 23no 5 pp 407ndash438 1973
[2] G D Cohen M G Karpovsky H F Mattson and J R SchatzldquoCovering radiusmdashsurvey and recent resultsrdquo IEEE Transac-tions on Information Theory vol 31 no 3 pp 328ndash343 1985
[3] G D Cohen S N Litsyn A C Lobstein and H F Mattson JrldquoCovering radius 1985ndash1994rdquoApplicable Algebra in EngineeringCommunications and Computing vol 8 no 3 pp 173ndash239 1997
[4] G D Cohen I Honkala S Litsyn and A Lobstein CoveringCodes Elsevier North-Holland The Netherlands 1997
[5] R L Graham and N J A Sloane ldquoOn the covering radius ofcodesrdquo IEEE Transactions on Information Theory vol 31 no 3pp 385ndash401 1985
[6] GD Cohen A C Lobstein andN J A Sloane ldquoFurther resultson the covering radius of codesrdquo IEEE Transactions on Informa-tion Theory vol 32 no 6 pp 680ndash694 1986
[7] AC Lobstein andG JM vanWee ldquoOnnormal and subnormalq-ary codesrdquo IEEE Transactions on Information Theory vol 35no 6 pp 1291ndash1295 1989
[8] R S Selvaraj and V Marka ldquoNormality of binary codes inRosenbloom-Tsfasman metricrdquo in Proceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo13) pp 1370ndash1373 August 2013
[9] M Yu Rosenbloom andMA Tsfasman ldquoCodes for them-met-ricrdquo Problems of Information Transmission vol 33 pp 45ndash521997
[10] M M Skriganov ldquoCoding theory and uniform distributionsrdquoSt Petersburg Mathematical Journal vol 143 no 2 pp 301ndash3372002
[11] I Siap and M Ozen ldquoThe complete weight enumerator forcodes over 119872
119899times119904 (119877) with respect to the Rosenbloom-Tsfasmanmetricrdquo Applied Mathematics Letters vol 17 no 1 pp 65ndash692004
[12] S TDougherty andK Shiromoto ldquoMaximumdistance codes in119872119886119905119899times119904
(Z119896)with a non-hammingmetric and uniformdistribu-
tionsrdquo Designs Codes and Cryptography vol 33 no 1 pp 45ndash61 2004
[13] S Jain ldquoCT burstsmdashfrom classical to array codingrdquo DiscreteMathematics vol 308 no 9 pp 1489ndash1499 2008
[14] B Yildiz I Siap T Bilgin and G Yesilot ldquoThe covering prob-lem for finite rings with respect to the RT-metricrdquo AppliedMathematics Letters vol 23 no 9 pp 988ndash992 2010
Now the above theorem can be restated in terms of 119897-cell in the following manner as the definition of the 119897-cellsuggests
Corollary 5 Let 119862 be an RT-metric code of length 119904 over F119902Then the covering radius of 119862 is 119904 minus 119897 if and only if exist an 119897-cellof the code 119862
Proposition 6 Let 1198621 and 1198622 be any two RT-metric codesover F119902 with parameters (1199041 1198701 1198891)1199021198771 and (1199042 1198702 1198892)1199021198772respectively Then their direct sum 119862 = 1198621⨁1198622 is an (1199041 +
Clearly this is a code of length 1199041 + 1199042 and cardinality 11987011198702Now since the minimum distance of 1198621 is 1198891 there exist twocodewords 1198881 and 119888
1015840
1in 1198621 such that 119889120588(1198881 119888
1015840
1) = 1198891 Then
the codewords 119888 = (1198881 1198882) and 1198881015840= (1198881015840
1 1198882) for some 1198882 isin 1198622
will also be at distance 1198891 which is minimum among all thecodewords of 119862 FromTheorem 4 and Definition 3 it is clearthat the covering radius of a code in RT-metric depends onthe partition number of the code and that the process ofpartitioning starts with the right most coordinate ThereforebyTheorem 4 unless1198622 is the space F
1199042119902 the partition number
of 119862 must be equal to that of 1198622 which is 1199042 minus 1198772 and hencethe covering radius of 119862 is 1199041 + 1199042 minus (1199042 minus 1198772) = 1199041 + 1198772
Remark 7 If 1198622 = F 1199042119902 then the covering radius of the code
119862 = 1198621⨁1198622 is 119877 = 1198771
4 Normality of Codes in RT-Metric
Proposition 8 If 119862 is an (119904 119870 119889120588)119902119877 RT-metric code over F119902then119873(119862) ge 119902119877
Proof The proof follows directly from the definition ofcovering radius and that of norm of a code
Theorem9 Any 119902-ary RT-metric code of length 119904 and coveringradius 119904 is normal and all the coordinates are acceptable
Proof As 119877 = 119904 the partition number is 0 which meansthat there exists an 120572 isin F119902 which is not present as the lastcoordinate of any codeword in 119862 If 119909 = (1199091 1199092 119909119904) isin F 119904
119902
is such that 119909119904 = 120572 then 119889120588(119909 119862(119894)
119886) = 119904 for each 119886 isin F119902
and for any 119894 isin 1 2 119904 Thus 119873(119894) = 119902119904 lt 119902119904 + 119902 minus 1 =
119902119877+(119902minus1) Hence the code is normal and all the coordinatesare acceptable
Lemma 10 Let 119862 be any RT-metric code with length 119904 andcovering radius 119877 lt 119904 then
119873(119894)
= (119902 minus 1) 119894 + 119877 (6)
for each 119894 isin 119877 + 1 119877 + 2 119904
Proof For a coordinate position 119894 isin 119877 + 1 119877 + 2 119904 thenorm is given by
119873(119894)
= max119909isinF 119904119902
sum
119886isinF119902
119889120588 (119909 119862(119894)
119886)
(7)
As the covering radius of 119862 is 119877 it has partition number119904 minus 119877 So the definition of partition number implies thateach codeword can be associated with a unique (119904 minus 119877)-tuplewhich actually consists of the last (119904 minus 119877) coordinates of thatcodeword and also that there exists at least one (119904 minus 119877 + 1)-tuple which is not the same as the last 119904 minus 119877 + 1 coordinatesof any of the codewords Now when we partition the codeinto 119902 parts 119862(119894)
119886 for 119886 isin F119902 we can also partition the set of
all (119904 minus 119877)-tuples into 119902 parts corresponding to the associatedcodewords in 119862
(119894)
119886 If we choose a word 119909 isin F 119904
119902whose last
119904 minus 119877 + 1 coordinates do not match with the last 119904 minus 119877 + 1
coordinates of any of the codewords then for this wordsum119886isinF119902
119889120588(119909 119862(119894)
119886) = (119902 minus 1)119894 + 119877 which is the maximum value
that a word can give Hence the proof holds
Theorem 11 Let 119862 be any RT-metric code of length 119904 withcovering radius 119877 lt 119904 Then the coordinates 119894 isin 119877 + 2 119877 +
3 119904 are not acceptable
Proof Theminimum norm is
119873 le 119873(119894) forall119894 isin 1 2 119904 (8)
By Lemma 10 119873(119894) = (119902 minus 1)119894 + 119877 for all 119894 ge 119877 + 1 whichimplies
The above theorem is not sufficient to arrive at a decisionon the acceptability of the 119904th coordinate when the codehas covering radius 119877 = 119904 minus 1 This can be settled by thefollowing theorem which says that no 119902-ary linear RT-metriccode with dimension more than 1 and covering radius 119904 minus 1
has acceptable last coordinate
Theorem 12 Let 119862 be an [119904 119896 119889120588]119902119877 linear RT-metric codeover F119902 with 119896 gt 1 and 119877 = 119904 minus 1 Then the last coordinate isnot acceptable
Proof By Lemma 10
119873(119904)
= (119902 minus 1) 119904 + 119877
= (119902 minus 1) 119904 + 119904 minus 1 = 119902119904 minus 1
= 119902 (119904 minus 1) + 119902 minus 1 = 119902119877 + (119902 minus 1)
(since 119877 = 119904 minus 1)
(11)
4 ISRN Combinatorics
In order to prove this theorem we must show that thereexists at least one coordinate 119895 for which 119873
(119895)lt 119873(119904) As
the dimension of 119862 is 119896 there will be 119896 coordinate positions1198941 1198942 119894119896 such that 119862
(119894119905)
119886= 0 for all 119886 isin F119902 and for each
119905 = 1 2 119896 One such coordinate position is 119904 (but not 119904minus1as the partition number is 1) As 119896 gt 1 there exists at leastonemore such coordinate position 119895 isin 1198941 1198942 119894119896 and 119895 = 119904
wherein 119862(119895)
119886= 0 for all 119886 isin F119902 and so 119873
(119895)le 119902(119904 minus 1) lt 119873
(119904)This completes the proof
But when the code is either of dimension 1 or of constantweight all the coordinates become acceptable as the followingresults suggest
Proposition 13 Let 119862 be an [119904 1]119902 linear RT-metric codeThen 119862 is normal and all the coordinates are acceptable
Proof It can easily be seen that the covering radius of 119862 iseither 119904 or 119904minus1 If it is 119904 the result follows byTheorem 9 If it is119904minus1 there exists a 1-cell Since119862 is of dimension 1 there exist119902 such 1-cells By Lemma 10 119873(119904) = 119902119904 minus 1 Now for each 119894 isin
1 2 119904minus1 either119862(119894)0
= 119862 or |119862(119894)0| = 1 In the former case
119862(119894)
119886= 0 for all 119886 = 0 and in the latter case |119862(119894)
119886| = 1 for all 119886 isin
F119902 In any case119873(119894)
= 119902119904minus1This proves the result stated
Proposition 14 Let 119862 be a constant weight code of length119904 over F119902 Then 119862 is normal and all the coordinates areacceptable
Proof It is easy to see that the covering radius 119877 of theconstant weight code 119862 is either 119904 or 119904 minus 1 as it can havepartition number of either 0 or 1 If119877 = 119904 then by Lemma 10it is done If 119877 = 119904 minus 1 then 119862
(119904)
119886= 0 for any 119886 isin F119902 Then for
any 119894 0 isin 119862(119894)
0and |119862
(119894)
0| ge 1 For an 119909 isin F 119904
119902with 119909119904 = 0 and
119909119904minus1 = 0119873(119894) equals 119902119904 minus 1 Hence the result follows
Remark 15 If 119862 = F 119904119902 then its partition number is 119904 and so its
covering radius 119877 is 0 By (3) for any 119894 119873(119894) = (119902 minus 1)119894 + 0 =
(119902 minus 1)119894 And so119873(1) lt 119873(2)
lt sdot sdot sdot lt 119873(119904) and119873
(1)= 119902 minus 1 =
119902119877+119902minus1 Thus the ambient space F 119904119902is normal with the first
coordinate being the only acceptable coordinate
41 Normality and Direct Sum Construction
Proposition 16 Let 1198621 and 1198622 be (1199041 1198701 1198891)1199021198771 and(1199042 1198702 1198892)1199021198772 RT-metric codes respectively such that 1198772 = 0Then
(i) the direct sum 119862 = 1198621⨁1198622 is normal(ii) all the coordinates corresponding to 1198621 are acceptable
for 119862(iii) the coordinate 119894 for 1199041 + 1 le 119894 le 1199042 is accept-
able for 119862 only if the norm of 1198622 with respect to thecoordinate 119894 minus 1199041 is equal to 1199021198772
Proof Let 1198621 and 1198622 be any (1199041 1198701 1198891)1199021198771 and (1199042
1198702 1198892)1199021198772 119902-ary RT-metric codes respectively Then by
Proposition 6 their direct sum 119862 = 1198621⨁1198622 is an (119904 11987011198702
1198891)119902119877 code with length 119904 = 1199041 + 1199042 and covering radius119877 = 1199041 + 1198772 Now for 119894 isin 1 2 1199041 by (3) we have
119873(119894)
(119862) = 119902 (1199041 + 1198772) (12)
Now in order to determine the norm 119873(119862) we have to find119873(119895)(119862) for each 119895 isin 1199041 +1 1199041 +2 1199041 + 1199042 the coordinate
positions corresponding to1198622 But we know by Proposition 8that119873(1198622) ge 1199021198772 which implies119873(119895)(119862) ge 119902(1199041+1198772) Hence119873(119862) = 119902(1199041+1198772) = 119902119877 lt 119902119877+119902minus1Thus119862 is normal and allthe coordinate positions corresponding to 1198621 are acceptableand the coordinate position 119895 isin 1199041 + 1 1199041 + 2 1199041 + 1199042
corresponding to 1198622 is acceptable if119873(119895minus1199041)(1198622) = 1199021198772
42 Normality and Amalgamated Direct Sum ConstructionThe amalgamated direct sum of two codes is defined asfollows
Definition 17 Let 1198621 be an [1199041 1198961]1199021198771 normal code with thelast coordinate being acceptable and let 1198622 be an [1199042 1198962]1199021198772normal code with the first coordinate being acceptableThentheir amalgamated direct sum (or shortly ADS) denoted by1198621⨁1198622 is an [1199041 + 1199042 minus 1 1198961 + 1198962 minus 1]119902(1199041 + 1198772 minus 1) code andis given by
Here in this definition we consider those codes 1198621 and 1198622which are normal respectively with the last coordinate andfirst coordinate being acceptable such that the parts 119862(119904)
1119886and
119862(1)
2119886are nonempty for all 119886 isin F119902
As Theorem 12 shows the nonexistence of linear codes ofdimension more than 1 whose last coordinate is acceptablethe significance of amalgamated direct sum and hence thatof normality in RT-metric are less as far as the resultingcovering radius is concerned But since the one-dimensionalcodes are normal and all the coordinates are acceptable onecan only use this ADS construction method to combine a1-dimensional code with any other linear code which maybe helpful in the construction of MDS codes as seen in thefollowing result
Theorem 18 Let 1198621 be an [1199041 1]119902 MDS code with coveringradius 119905119902[1199041 1] (which is normal with last coordinate beingacceptable) and let 1198622 be an [1199042 1198962]119902 MDS code with coveringradius 119905119902[1199042 1198962] Then their amalgamated direct sum 1198621⨁1198622is an [1199041 + 1199042 minus 1 1198962]119902 MDS code with covering radius 119905119902[1199041 +1199042 minus 1 1198962]
Proof From the definition of partition number and fromTheorem 4 it is easy to see that 119905119902[119904 119896] = 119904 minus 119896 So 119905119902[1199041 1] =1199041 minus 1 and 119905119902[1199042 1198962] = 1199042 minus 1198962 Now as the dimension of1198621⨁1198622 is 1198962 (which is less than or equal to 1199042) the coveringradius of1198621⨁1198622 depends only on the coordinates pertainingto the code 1198622 But 1198622 has covering radius 1199042 minus 1198962 and hence
ISRN Combinatorics 5
has partition number 1198962 So 1198621⨁1198622 also will have partitionnumber 1198962 and hence will have covering radius 1199041+1199042minus1minus1198962
which is equal to 119905119902[1199041+1199042minus1 1198962] which implies that1198621⨁1198622is MDS This completes the proof
5 Conclusion
We discussed the normality of codes over F119902 in RT-metricand determined its norm with respect to various coordinatepositions We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Delsarte ldquoFour fundamental parameters of a code and theircombinatorial significancerdquo Information and Control vol 23no 5 pp 407ndash438 1973
[2] G D Cohen M G Karpovsky H F Mattson and J R SchatzldquoCovering radiusmdashsurvey and recent resultsrdquo IEEE Transac-tions on Information Theory vol 31 no 3 pp 328ndash343 1985
[3] G D Cohen S N Litsyn A C Lobstein and H F Mattson JrldquoCovering radius 1985ndash1994rdquoApplicable Algebra in EngineeringCommunications and Computing vol 8 no 3 pp 173ndash239 1997
[4] G D Cohen I Honkala S Litsyn and A Lobstein CoveringCodes Elsevier North-Holland The Netherlands 1997
[5] R L Graham and N J A Sloane ldquoOn the covering radius ofcodesrdquo IEEE Transactions on Information Theory vol 31 no 3pp 385ndash401 1985
[6] GD Cohen A C Lobstein andN J A Sloane ldquoFurther resultson the covering radius of codesrdquo IEEE Transactions on Informa-tion Theory vol 32 no 6 pp 680ndash694 1986
[7] AC Lobstein andG JM vanWee ldquoOnnormal and subnormalq-ary codesrdquo IEEE Transactions on Information Theory vol 35no 6 pp 1291ndash1295 1989
[8] R S Selvaraj and V Marka ldquoNormality of binary codes inRosenbloom-Tsfasman metricrdquo in Proceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo13) pp 1370ndash1373 August 2013
[9] M Yu Rosenbloom andMA Tsfasman ldquoCodes for them-met-ricrdquo Problems of Information Transmission vol 33 pp 45ndash521997
[10] M M Skriganov ldquoCoding theory and uniform distributionsrdquoSt Petersburg Mathematical Journal vol 143 no 2 pp 301ndash3372002
[11] I Siap and M Ozen ldquoThe complete weight enumerator forcodes over 119872
119899times119904 (119877) with respect to the Rosenbloom-Tsfasmanmetricrdquo Applied Mathematics Letters vol 17 no 1 pp 65ndash692004
[12] S TDougherty andK Shiromoto ldquoMaximumdistance codes in119872119886119905119899times119904
(Z119896)with a non-hammingmetric and uniformdistribu-
tionsrdquo Designs Codes and Cryptography vol 33 no 1 pp 45ndash61 2004
[13] S Jain ldquoCT burstsmdashfrom classical to array codingrdquo DiscreteMathematics vol 308 no 9 pp 1489ndash1499 2008
[14] B Yildiz I Siap T Bilgin and G Yesilot ldquoThe covering prob-lem for finite rings with respect to the RT-metricrdquo AppliedMathematics Letters vol 23 no 9 pp 988ndash992 2010
In order to prove this theorem we must show that thereexists at least one coordinate 119895 for which 119873
(119895)lt 119873(119904) As
the dimension of 119862 is 119896 there will be 119896 coordinate positions1198941 1198942 119894119896 such that 119862
(119894119905)
119886= 0 for all 119886 isin F119902 and for each
119905 = 1 2 119896 One such coordinate position is 119904 (but not 119904minus1as the partition number is 1) As 119896 gt 1 there exists at leastonemore such coordinate position 119895 isin 1198941 1198942 119894119896 and 119895 = 119904
wherein 119862(119895)
119886= 0 for all 119886 isin F119902 and so 119873
(119895)le 119902(119904 minus 1) lt 119873
(119904)This completes the proof
But when the code is either of dimension 1 or of constantweight all the coordinates become acceptable as the followingresults suggest
Proposition 13 Let 119862 be an [119904 1]119902 linear RT-metric codeThen 119862 is normal and all the coordinates are acceptable
Proof It can easily be seen that the covering radius of 119862 iseither 119904 or 119904minus1 If it is 119904 the result follows byTheorem 9 If it is119904minus1 there exists a 1-cell Since119862 is of dimension 1 there exist119902 such 1-cells By Lemma 10 119873(119904) = 119902119904 minus 1 Now for each 119894 isin
1 2 119904minus1 either119862(119894)0
= 119862 or |119862(119894)0| = 1 In the former case
119862(119894)
119886= 0 for all 119886 = 0 and in the latter case |119862(119894)
119886| = 1 for all 119886 isin
F119902 In any case119873(119894)
= 119902119904minus1This proves the result stated
Proposition 14 Let 119862 be a constant weight code of length119904 over F119902 Then 119862 is normal and all the coordinates areacceptable
Proof It is easy to see that the covering radius 119877 of theconstant weight code 119862 is either 119904 or 119904 minus 1 as it can havepartition number of either 0 or 1 If119877 = 119904 then by Lemma 10it is done If 119877 = 119904 minus 1 then 119862
(119904)
119886= 0 for any 119886 isin F119902 Then for
any 119894 0 isin 119862(119894)
0and |119862
(119894)
0| ge 1 For an 119909 isin F 119904
119902with 119909119904 = 0 and
119909119904minus1 = 0119873(119894) equals 119902119904 minus 1 Hence the result follows
Remark 15 If 119862 = F 119904119902 then its partition number is 119904 and so its
covering radius 119877 is 0 By (3) for any 119894 119873(119894) = (119902 minus 1)119894 + 0 =
(119902 minus 1)119894 And so119873(1) lt 119873(2)
lt sdot sdot sdot lt 119873(119904) and119873
(1)= 119902 minus 1 =
119902119877+119902minus1 Thus the ambient space F 119904119902is normal with the first
coordinate being the only acceptable coordinate
41 Normality and Direct Sum Construction
Proposition 16 Let 1198621 and 1198622 be (1199041 1198701 1198891)1199021198771 and(1199042 1198702 1198892)1199021198772 RT-metric codes respectively such that 1198772 = 0Then
(i) the direct sum 119862 = 1198621⨁1198622 is normal(ii) all the coordinates corresponding to 1198621 are acceptable
for 119862(iii) the coordinate 119894 for 1199041 + 1 le 119894 le 1199042 is accept-
able for 119862 only if the norm of 1198622 with respect to thecoordinate 119894 minus 1199041 is equal to 1199021198772
Proof Let 1198621 and 1198622 be any (1199041 1198701 1198891)1199021198771 and (1199042
1198702 1198892)1199021198772 119902-ary RT-metric codes respectively Then by
Proposition 6 their direct sum 119862 = 1198621⨁1198622 is an (119904 11987011198702
1198891)119902119877 code with length 119904 = 1199041 + 1199042 and covering radius119877 = 1199041 + 1198772 Now for 119894 isin 1 2 1199041 by (3) we have
119873(119894)
(119862) = 119902 (1199041 + 1198772) (12)
Now in order to determine the norm 119873(119862) we have to find119873(119895)(119862) for each 119895 isin 1199041 +1 1199041 +2 1199041 + 1199042 the coordinate
positions corresponding to1198622 But we know by Proposition 8that119873(1198622) ge 1199021198772 which implies119873(119895)(119862) ge 119902(1199041+1198772) Hence119873(119862) = 119902(1199041+1198772) = 119902119877 lt 119902119877+119902minus1Thus119862 is normal and allthe coordinate positions corresponding to 1198621 are acceptableand the coordinate position 119895 isin 1199041 + 1 1199041 + 2 1199041 + 1199042
corresponding to 1198622 is acceptable if119873(119895minus1199041)(1198622) = 1199021198772
42 Normality and Amalgamated Direct Sum ConstructionThe amalgamated direct sum of two codes is defined asfollows
Definition 17 Let 1198621 be an [1199041 1198961]1199021198771 normal code with thelast coordinate being acceptable and let 1198622 be an [1199042 1198962]1199021198772normal code with the first coordinate being acceptableThentheir amalgamated direct sum (or shortly ADS) denoted by1198621⨁1198622 is an [1199041 + 1199042 minus 1 1198961 + 1198962 minus 1]119902(1199041 + 1198772 minus 1) code andis given by
Here in this definition we consider those codes 1198621 and 1198622which are normal respectively with the last coordinate andfirst coordinate being acceptable such that the parts 119862(119904)
1119886and
119862(1)
2119886are nonempty for all 119886 isin F119902
As Theorem 12 shows the nonexistence of linear codes ofdimension more than 1 whose last coordinate is acceptablethe significance of amalgamated direct sum and hence thatof normality in RT-metric are less as far as the resultingcovering radius is concerned But since the one-dimensionalcodes are normal and all the coordinates are acceptable onecan only use this ADS construction method to combine a1-dimensional code with any other linear code which maybe helpful in the construction of MDS codes as seen in thefollowing result
Theorem 18 Let 1198621 be an [1199041 1]119902 MDS code with coveringradius 119905119902[1199041 1] (which is normal with last coordinate beingacceptable) and let 1198622 be an [1199042 1198962]119902 MDS code with coveringradius 119905119902[1199042 1198962] Then their amalgamated direct sum 1198621⨁1198622is an [1199041 + 1199042 minus 1 1198962]119902 MDS code with covering radius 119905119902[1199041 +1199042 minus 1 1198962]
Proof From the definition of partition number and fromTheorem 4 it is easy to see that 119905119902[119904 119896] = 119904 minus 119896 So 119905119902[1199041 1] =1199041 minus 1 and 119905119902[1199042 1198962] = 1199042 minus 1198962 Now as the dimension of1198621⨁1198622 is 1198962 (which is less than or equal to 1199042) the coveringradius of1198621⨁1198622 depends only on the coordinates pertainingto the code 1198622 But 1198622 has covering radius 1199042 minus 1198962 and hence
ISRN Combinatorics 5
has partition number 1198962 So 1198621⨁1198622 also will have partitionnumber 1198962 and hence will have covering radius 1199041+1199042minus1minus1198962
which is equal to 119905119902[1199041+1199042minus1 1198962] which implies that1198621⨁1198622is MDS This completes the proof
5 Conclusion
We discussed the normality of codes over F119902 in RT-metricand determined its norm with respect to various coordinatepositions We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Delsarte ldquoFour fundamental parameters of a code and theircombinatorial significancerdquo Information and Control vol 23no 5 pp 407ndash438 1973
[2] G D Cohen M G Karpovsky H F Mattson and J R SchatzldquoCovering radiusmdashsurvey and recent resultsrdquo IEEE Transac-tions on Information Theory vol 31 no 3 pp 328ndash343 1985
[3] G D Cohen S N Litsyn A C Lobstein and H F Mattson JrldquoCovering radius 1985ndash1994rdquoApplicable Algebra in EngineeringCommunications and Computing vol 8 no 3 pp 173ndash239 1997
[4] G D Cohen I Honkala S Litsyn and A Lobstein CoveringCodes Elsevier North-Holland The Netherlands 1997
[5] R L Graham and N J A Sloane ldquoOn the covering radius ofcodesrdquo IEEE Transactions on Information Theory vol 31 no 3pp 385ndash401 1985
[6] GD Cohen A C Lobstein andN J A Sloane ldquoFurther resultson the covering radius of codesrdquo IEEE Transactions on Informa-tion Theory vol 32 no 6 pp 680ndash694 1986
[7] AC Lobstein andG JM vanWee ldquoOnnormal and subnormalq-ary codesrdquo IEEE Transactions on Information Theory vol 35no 6 pp 1291ndash1295 1989
[8] R S Selvaraj and V Marka ldquoNormality of binary codes inRosenbloom-Tsfasman metricrdquo in Proceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo13) pp 1370ndash1373 August 2013
[9] M Yu Rosenbloom andMA Tsfasman ldquoCodes for them-met-ricrdquo Problems of Information Transmission vol 33 pp 45ndash521997
[10] M M Skriganov ldquoCoding theory and uniform distributionsrdquoSt Petersburg Mathematical Journal vol 143 no 2 pp 301ndash3372002
[11] I Siap and M Ozen ldquoThe complete weight enumerator forcodes over 119872
119899times119904 (119877) with respect to the Rosenbloom-Tsfasmanmetricrdquo Applied Mathematics Letters vol 17 no 1 pp 65ndash692004
[12] S TDougherty andK Shiromoto ldquoMaximumdistance codes in119872119886119905119899times119904
(Z119896)with a non-hammingmetric and uniformdistribu-
tionsrdquo Designs Codes and Cryptography vol 33 no 1 pp 45ndash61 2004
[13] S Jain ldquoCT burstsmdashfrom classical to array codingrdquo DiscreteMathematics vol 308 no 9 pp 1489ndash1499 2008
[14] B Yildiz I Siap T Bilgin and G Yesilot ldquoThe covering prob-lem for finite rings with respect to the RT-metricrdquo AppliedMathematics Letters vol 23 no 9 pp 988ndash992 2010
has partition number 1198962 So 1198621⨁1198622 also will have partitionnumber 1198962 and hence will have covering radius 1199041+1199042minus1minus1198962
which is equal to 119905119902[1199041+1199042minus1 1198962] which implies that1198621⨁1198622is MDS This completes the proof
5 Conclusion
We discussed the normality of codes over F119902 in RT-metricand determined its norm with respect to various coordinatepositions We also established that the last coordinate is notacceptable for any nontrivial code in this metric whichmakesthe ADS construction less significant as far as the higherdimensional RT-metric codes are concerned
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Delsarte ldquoFour fundamental parameters of a code and theircombinatorial significancerdquo Information and Control vol 23no 5 pp 407ndash438 1973
[2] G D Cohen M G Karpovsky H F Mattson and J R SchatzldquoCovering radiusmdashsurvey and recent resultsrdquo IEEE Transac-tions on Information Theory vol 31 no 3 pp 328ndash343 1985
[3] G D Cohen S N Litsyn A C Lobstein and H F Mattson JrldquoCovering radius 1985ndash1994rdquoApplicable Algebra in EngineeringCommunications and Computing vol 8 no 3 pp 173ndash239 1997
[4] G D Cohen I Honkala S Litsyn and A Lobstein CoveringCodes Elsevier North-Holland The Netherlands 1997
[5] R L Graham and N J A Sloane ldquoOn the covering radius ofcodesrdquo IEEE Transactions on Information Theory vol 31 no 3pp 385ndash401 1985
[6] GD Cohen A C Lobstein andN J A Sloane ldquoFurther resultson the covering radius of codesrdquo IEEE Transactions on Informa-tion Theory vol 32 no 6 pp 680ndash694 1986
[7] AC Lobstein andG JM vanWee ldquoOnnormal and subnormalq-ary codesrdquo IEEE Transactions on Information Theory vol 35no 6 pp 1291ndash1295 1989
[8] R S Selvaraj and V Marka ldquoNormality of binary codes inRosenbloom-Tsfasman metricrdquo in Proceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo13) pp 1370ndash1373 August 2013
[9] M Yu Rosenbloom andMA Tsfasman ldquoCodes for them-met-ricrdquo Problems of Information Transmission vol 33 pp 45ndash521997
[10] M M Skriganov ldquoCoding theory and uniform distributionsrdquoSt Petersburg Mathematical Journal vol 143 no 2 pp 301ndash3372002
[11] I Siap and M Ozen ldquoThe complete weight enumerator forcodes over 119872
119899times119904 (119877) with respect to the Rosenbloom-Tsfasmanmetricrdquo Applied Mathematics Letters vol 17 no 1 pp 65ndash692004
[12] S TDougherty andK Shiromoto ldquoMaximumdistance codes in119872119886119905119899times119904
(Z119896)with a non-hammingmetric and uniformdistribu-
tionsrdquo Designs Codes and Cryptography vol 33 no 1 pp 45ndash61 2004
[13] S Jain ldquoCT burstsmdashfrom classical to array codingrdquo DiscreteMathematics vol 308 no 9 pp 1489ndash1499 2008
[14] B Yildiz I Siap T Bilgin and G Yesilot ldquoThe covering prob-lem for finite rings with respect to the RT-metricrdquo AppliedMathematics Letters vol 23 no 9 pp 988ndash992 2010
