1004 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013 Codes … ·...

1004 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013

Codes and Designs Related to Lifted MRD CodesTuvi Etzion, Fellow, IEEE, and Natalia Silberstein

Abstract—Lifted maximum rank distance (MRD) codes, whichare constant dimension codes, are considered. It is shown that alifted MRD code can be represented in such a way that it formsa block design known as a transversal design. A slightly differentrepresentation of this design makes it similar to a -analog of atransversal design. The structure of these designs is used to obtainupper bounds on the sizes of constant dimension codes which con-tain a lifted MRD code. Codes that attain these bounds are con-structed. These codes are the largest known constant dimensioncodes for the given parameters. These transversal designs can alsobe used to derive a new family of linear codes in the Hammingspace. Bounds on the minimum distance and the dimension of suchcodes are given.

Index Terms—Constant dimension codes, Grassmannian space,lifted maximum rank distance (MRD) codes, rank-metric codes,transversal designs.

I. INTRODUCTION

L ET be the finite field of size . For two matricesand over , the rank distance is defined by

A rank-metric code is a linear code, whose code-words are matrices over ; they form a linear subspacewith dimension of , and for each two distinct codewordsand , we have that . For a rank-

metric code , it was proved in [10], [17], [35] that

(1)

This bound, called Singleton bound for the rank metric, is at-tained for all feasible parameters. The codes which attain thisbound are called maximum rank distance codes (or MRD codesin short).Rank-metric codes have found application in public key

cryptosystems [18], space-time coding [32], authenticationcodes [52], rank-minimization over finite fields [44], anddistributed storage systems [41]. Recently, rank-metric codesalso have found a new application in the construction oferror-correcting codes for random network coding [42]. For

Manuscript received September 05, 2011; revised July 26, 2012; acceptedSeptember 11, 2012. Date of publication October 22, 2012; date of current ver-sion January 16, 2013. This work was supported in part by the Israel ScienceFoundation, Jerusalem, Israel, under Grant 230/08. This paper was presented inpart at the 2011 IEEE International Symposium on Information Theory.T. Etzion is with the Department of Computer Science, Technion—Israel In-

stitute of Technology, Haifa 32000, Israel (e-mail: [email protected]).N. Silberstein is with the Department of Electrical and Computer Engi-

neering, University of Texas at Austin, Austin, TX 78712-1684 USA (e-mail:[email protected]).Communicated by D. Burshtein, Associate Editor for Coding Techniques.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2012.2220119

this application, the matrices are lifted into -dimensionalsubspaces of [42] as described in the following.Let be a matrix over and let be a identity

matrix. The matrix can be viewed as a generator matrixof a -dimensional subspace of , and it is called the liftingof [42].

Example 1: Let and be the following matrices over:

Then, the subspace obtained by the lifting of is given by thefollowing eight vectors:

Given a nonnegative integer , the set of all -dimensionalsubspaces of forms theGrassmannian space (Grassmannianin short) over , which is denoted by . It is well knownthat , where is the-ary Gaussian coefficient.A subset of is called an constant

dimension code if it has size and minimum subspace distance, where the distance function in is defined by

for any two subspaces and in . willdenote the maximum size of an code.Codes in the Grassmannian gained recently lot of interest

due to the work by Koetter and Kschischang [26], where theypresented an application of such codes for error correction inrandom network coding. When the codewords of a rank-metriccode are lifted to -dimensional subspaces, the result is a con-stant dimension code . If is an MRD code, then is calleda lifted MRD code [42]. This code will be denoted by .

Theorem 1: [42]: Let and be positive integers such that. If is a MRD

code, then is an code.In view of Theorem 1, we will assume throughout

this paper that . which is ancode will be also called an

. If no parameters for will be given, wewill assume it is an .Most of the constructions for large constant dimension codes

known in the literature produce codes which contain[13], [20], [33], [39], [42], [43], [48]. The only constructionswhich generate codes that do not contain are given in[15], [27] and [49]. These constructions are either of so-called

0018-9448/$31.00 © 2012 IEEE

ETZION AND SILBERSTEIN: CODES AND DESIGNS RELATED TO LIFTED MRD CODES 1005

orbit codes or specific constructions for small parameters.More-over, only orbit codes (specifically cyclic codes)with , and and codes arethe largest codes for their specific parameters which do not con-tain [27]. This motivates the question, what is the largestconstant dimension code which contains ?The well-known concept of -analogs replaces subsets by

subspaces of a vector space over a finite field and their ordersby the dimensions of the subspaces. In particular, the -analogof a constant weight code in the Johnson space is a constant di-mension code in the Grassmannian space. Related to constantdimension codes are -analogs of block designs. -analogs ofdesigns were studied in [1], [7], [15], [16], [37] and [47]. For ex-ample, in [1], it was shown that Steiner structures (the -analogof Steiner systems), if exist, yield optimal codes in the Grass-mannian. Another connection is the constructions of constantdimension codes from spreads which are given in [15] and [33].In this paper, we consider several topics related to lifted

MRD codes. First, we discuss properties of these codes relatedto block designs. We prove that the codewords of forma design called a transversal design, a structure which is knownto be equivalent to the well-known orthogonal array. We alsoprove that the same codewords form a subspace transversaldesign, which is akin to the transversal design, but not its-analog.The structure of as a transversal design leads to

the other results given in this paper. We derive for newlower bounds on and upper bounds on the sizesof error-correcting constant dimension codes which contain

. In particular, we prove that if ancode , , contains an code, then

We present a construction for codes that either attain thisbound or almost attain it for . These codes are the largestknown codes for .We prove that if an code contains an

code, then

We present a construction for codes that attain this boundwhen , , and for all . These codes are the largestknown for the related parameters.The incidence matrix of the transversal design derived from

can be viewed as a parity-check matrix of a linear code inthe Hamming space. This way to construct a linear code from adesign is well known [2], [12], [23], [25], [28]–[30], [50], [51],[55]. We find the properties of these codes, in particular, wepresent the bounds on their minimum distance and dimension.The rest of this paper is organized as follows. In Section II,

we present properties of lifted MRD codes. Then, we provethat these codes form transversal designs in sets and subspaces.In Section III, we discuss some known upper bounds on

and present two new upper bounds on the sizes ofconstant dimension codes which contain . In Sections IVand V, we provide constructions of two families of codes

that attain the upper bounds of Section III. In Section VI weconsider properties of linear codes whose parity-check matricesare derived from . Conclusions and problems for futureresearch are given in Section VII.

II. LIFTED MRD CODES AND TRANSVERSAL DESIGNS

In this section, we prove that a liftedMRD code yields a com-binatorial structure known as a transversal design. Moreover,the codewords of these codes form the blocks of a new type oftransversal design, called a subspace transversal design. Basedon these designs, we will present some novel results in the fol-lowing sections. We first examine some combinatorial proper-ties of lifted MRD codes. Based on these properties, we willconstruct the transversal designs.

A. Properties of Lifted MRD Codes

Let be the set of vectors of length overin which not all the first entries are zeroes. The following

lemma is a simple observation.

Lemma 2: All the nonzero vectors that are contained in code-words of an belong to .For a set , let denote the subspace of spanned

by the elements of . If is of size one, then we denoteby . For and , we denote by

the concatenation of and . Let

be the set of all one-dimensional subspaces ofwhose nonzero vectors are contained in . We identify

each subspace , for any given , with the vector(of length ) in which the first nonzero entry is a one.

For each , we define

in other words, consists of all one-dimensional subspaceswhose restriction to the first coordinates is precisely .

contains sets, each one of thesize . These sets partition the set , i.e.,

and

We say that a vector is in if for

. Clearly, , for and, contains a vector with leading zeroes. Such a vector

does not belong to , and hence, by Lemma 2, we havethe following.

Lemma 3: For each , a codeword ofcontains at most one element from .Note that each -dimensional subspace of contains

one-dimensional subspaces. Therefore, by

Lemma 2, each codeword of contains elements

of . Hence, by Lemma 3, and since , wehave the following.


Corollary 1: For each , a codeword ofcontains exactly one element from .

Lemma 4: Each -dimensional subspace of ,whose nonzero vectors are contained in , is contained inexactly one codeword of an .

Proof: Let

i.e., consists of all -dimensional subspaces ofin which all the nonzero vectors are contained in .Since theminimum distance of is and its codewords

are -dimensional subspaces, it follows that the intersection ofany two codewords is at most of dimension . Hence, each

-dimensional subspace of is contained in at mostone codeword. The size of is , and thenumber of -dimensional subspaces in a codeword

is exactly . By Lemma 2, each -dimen-

sional subspace, of a codeword, is contained in . Hence, thecodewords of contain exactly

distinct -dimensional subspaces of .To complete the proof, we only have to show that does not

contain more -dimensional subspaces. Hence, wewill compute the size of . Each element of intersects witheach , in at most one one-dimensional sub-space (since it contains vectors only from ). There are

ways to choose an arbitrary -dimensional

subspace of . For each such subspace, we choose a basis, where each belongs to a different set

, (clearly, by previous definition, in eachsuch basis vector, the first nonzero entry is a one).A basis for a -dimensional subspace of will be

generated by concatenation of with a vector foreach , .Therefore, there are ways to choose a basis for

an element of . Hence, .

Thus, the lemma follows.

Corollary 2: For each , , each-dimensional subspace of , whose nonzero vec-

tors are contained in , is contained in exactlycodewords of .

Proof: The size of is . The numberof -dimensional subspaces in a codeword is exactly

. Hence, the total number of -dimensional

subspaces in is (clearly, each

such -dimensional subspace is counted more thanonce in this computation). Similarly to the proof of Lemma 4,we can prove that the total number of -dimensionalsubspaces that contain nonzero vectors only from is

. By simple symmetry, each two dif-

ferent such subspaces, containing nonzero vectors only from, are contained in the same number of codewords of. Thus, each -dimensional subspace of ,

whose nonzero vectors are contained in , is contained inexactly

codewords of .

Corollary 3: Each one-dimensional subspace iscontained in exactly codewords of .By applying Corollary 2 with , we also infer the

following result.Corollary 4: Any two elements , such that

and , , are contained in ex-actly codewords of .For the following lemma, we need a generalization of the

definition of a rank-metric code to a nonlinear rank-metric code,which is a subset of with minimum distance and size .If , then such a code willbe also called an MRD code.

Lemma 5: can be partitioned into sets,called parallel classes, each one of size , such that in eachparallel class, each element of is contained in exactly onecodeword.

Proof: First, we prove that a lifted MRD code contains alifted MRD subcode with disjoint codewords (subspaces). Letbe the generator matrix of a

MRD code [17], . Then, has the following form:

......

...

where are linearly independent over . If the lastrows are removed from , the result is an MRD sub-

code of with the minimum distance . In other words, anMRD subcode of is obtained. The

corresponding lifted code is an lifted MRDsubcode of .Let be the cosets

of in . All these cosets are nonlinear rank-metriccodes with the same parameters as theMRD code. Therefore, their lifted codes form a partition of an

into parallel classes each one ofsize , such that each element of is contained in exactlyone codeword of each parallel class.

B. Transversal Designs From Lifted MRD Codes

A transversal design of groupsize , blocksize , strength ,and index , denoted by is a triple ,where:1) is a set of elements (called points);2) is a partition of into classes (called groups), eachone of size ;

3) is a collection of -subsets of (called blocks);4) each block meets each group in exactly one point;


5) each -subset of points that meets each group in at mostone point is contained in exactly blocks.

When , the strength is usually not mentioned, and thedesign is denoted by . A is resolvableif the set can be partitioned into sets , where eachelement of is contained in exactly one block of each . Thesets are called parallel classes.

Example 2: Let ; ,where , , and

; , where, , , ,

, , ,, , ,, , ,, , and .

These blocks form a resolvable with four parallelclasses , ,

, and .

Theorem 6: The codewords of an form the

blocks of a resolvable transversal design ,, with parallel classes, each one

of size .Proof: Let be the set of points for the design.

Each set , , is defined to be a group, i.e., there

are groups, each one of size . The -dimensional sub-spaces (codewords) of are the blocks of the design. ByCorollary 1, each block meets each group in exactly one point.By Corollary 4, each 2-subset whichmeets each group in at mostone point is contained in exactly blocks. Finally,by Lemma 5, the design is resolvable with parallelclasses, each one of size .

An array with entries from a set of elementsis an orthogonal array with levels, strength , and index ,denoted by , if every subarray ofcontains each -tuple exactly times as a row. It is known[21] that a is equivalent to an orthogonal array

.A MRD code is

a maximum distance separable (MDS) code if it is viewedas a code of length over [17]. Thus, its codewordsform an orthogonal array

with , which is also an orthogonal arraywith (see

[21] for the connection between MDS codes and orthogonalarrays).By the equivalence of transversal designs and orthogonal

arrays, and by Theorem 6, an code induces an

with .These parameters are different from the ones obtained byviewing an MRD code as an MDS code.Now, we define a new type of transversal designs in terms of

subspaces, which will be called a subspace transversal design.We will show that such a design is induced by the codewords ofa lifted MRD code. Moreover, we will show that this design isuseful to obtain upper bounds on the codes that contain the lifted

MRD codes, and in a construction of large constant dimensioncodes.Let be a set of one-dimensional subspaces in

that contains only vectors starting with zeroes. Note thatis isomorphic to .

A subspace transversal design of groupsize , ,block dimension , and strength , denoted by ,is a triple , where1) is the subset of all elements of ,

(the points);

2) is a partition of into classes of size (thegroups);

3) is a collection of -dimensional subspaces which containonly points from (the blocks);

4) each block meets each group in exactly one point;5) each -dimensional subspace (with points from ) whichmeets each group in at most one point is contained in ex-actly one block.

An is resolvable if the set can be parti-tioned into sets , where each one-dimensional sub-space of is contained in exactly one block of each . Thesets are called parallel classes.As a direct consequence from Lemma 4 and Theorem 6, we

infer the following theorem.

Theorem 7: The codewords of an form theblocks of a resolvable , with the set ofpoints and the set of groups , , definedpreviously in this section.

Remark 1: There is no known nontrivial -analog of a blockdesign with and . An is very closeto such a design.

Remark 2: An cannot exist if ,unless . This is not difficult to prove and we leave it as anexercise for the interested reader. Recall that the casewas not considered in this section (see Theorem 1).

III. UPPER BOUNDS ON THE SIZE OF CODES IN

In this section, we consider upper bounds on the size of con-stant dimension codes. First, in Section III-A, we consider theJohnson type upper bound presented in [14], [15], [52] and [53].We estimate the size of known constant dimension codes rela-tively to this bound. The estimations provide better results thanthe ones known before, e.g., [26]. In Section III-B, we providenew upper bounds on codes that contain lifted MRD codes. Thistype of upper bounds was not considered before, even so, as saidearlier, usually the largest known codes contain the lifted MRDcodes.

A. Some Known Upper Bounds

Upper bounds on the sizes of constant dimension codes wereobtained in several papers, e.g., [26] and [42]. The followingupper bound was established in [52] in the context of linear au-thentication codes and in [14], [15] and [53] based on anticodesin the Grassmannian and as generalization of the well-knownJohnson bound for constant weight codes.


TABLE I

Theorem 8:

(2)

It was proved recently [6] that for fixed , , and , the ratiobetween the upper bound of Theorem 8 and equals 1as . But the method used in [6] is based on probabilisticarguments and an explicit construction of the related code is notknown. We will estimate the value of this upper bound

We define , . Similar analysis forwas considered in [26] and was considered also in

[19]. Since has codewords, wehave the following.

Lemma 9: The ratio between the size of anand the upper bound on given in (2) satisfies

The function is increasing in and also in . In Table I,we provide several values of for different and . For

, these values were given in [4].One can verify that for large enough or for large enough,

the size of a lifted MRD code approaches the upper bound (2).Thus, an improvement on the lower bound of ismainly important for small minimum distance and small . Thiswill be the line of research in the following sections.Note that the lower bound of Lemma 9 is not precise for small

values of . But it is better improved by another construction,the multilevel construction [13]. For example, for , thelower bound on the ratio between the size of a constant dimen-sion code generated by the multilevel construction andthe upper bound on given in (2) is presented inTable II. The values in the table are larger than the related valuesin Table I. In the construction of such a code , we consideronly code and the codewords related to the followingthree identifying vectors (see [13] or Section IV for the defi-nitions) , , and

TABLE IILOWER BOUNDS ON RATIO BETWEEN AND THE BOUND IN (2)

, which constitute most of the code.

But since not all identifying vectors were taken in the computa-tions, the values in Table II are only lower bounds on the ratio,rather than the exact ratio.

B. Upper Bounds for Codes Which Contain Lifted MRD Codes

In this section, we will derive upper bounds on the size of aconstant dimension code which contains the lifted MRD code

.Let be a subspace transversal design derived from

by Theorem 7. Recall that is the setof vectors of length over in which not all thefirst entries are zeroes. Let be the set of vectors in

which start with zeroes. is isomorphic to ,

, and . Note that isthe set of one-dimensional subspaces of that containonly vectors from . A codeword of a constant dimensioncode, in , contains one-dimensional subspaces from

. Let be a constant dimension codesuch that . Each codeword of containseither at least two points from the same group of or onlypoints from , and hence, it contains vectors of .

Theorem 10: If an code , , con-tains an , then

.Proof: Let be an obtained from an

. Since the minimum distance ofis , it follows that any two codewords of intersectin at most one one-dimensional subspace. Hence, each two-di-mensional subspace of is contained in at most one code-word of . Each two-dimensional subspace of , such that

, , , where ,, is contained in a codeword of by Corollary 4

(or by Theorem 7). Hence, each codeword ei-ther contains only points from or contains points from

and points from , for some . Clearly,in the first case and

in the second case. Since and two codewords ofintersect in at most one-dimensional subspace, it follows that

each -dimensional subspace of can be containedonly in one codeword. Moreover, since the minimum distanceof the code is , it follows that ifand , then

. Therefore,is an

code. Let be the set of code-words in such that . For each


, let be an arbitrary -dimensional subspace of

, and let (note that ). Since, , and each two codewords of in-

tersect in at most one-dimensional subspace, it follows that thecode is an code. Thisimplies the result of the theorem.

Theorem 11: If an code con-tains an , then

.

Proof: Let be an obtained froman . Since the minimum distance ofis , it follows that any two codewords of intersect in atmost a -dimensional subspace. Hence, each -dimen-sional subspace of is contained in at most one codewordof . Each -dimensional subspace of , such that

, , where ,for , and , , is con-tained in a codeword of by Theorem 7. Hence, eachcodeword has a nonempty intersection withexactly groups of , for some , and there-

fore, . Let be the set of codewordsdefined by if .The set forms an code, and hence,

.Let be a -dimensional subspace of . If and

are two codewords that contain , then . Letbe the number of codewords from which contain . Clearly,for each , , we have

(3)

There are points in and each con-

tains exactly points from . Hence, each -dimen-

sional subspace of can be a subspace of at most

codewords of .Therefore

where the equality is derived from (3).One can easily verify that for

; recall also that ;thus, we have

IV. CONSTRUCTIONS FOR CODES

In this section, we discuss and present a construction of codeswhich contain an and attain the bound of The-orem 10. Such a construction is presented only for andlarge enough. If is not large enough, then codes obtained bya modification of this construction almost attain the bound. Inany case, the codes obtained in this section are the largest onesknown for and .For , the upper bound of Theorem 10 on the size of

a code that contains an is .The construction which follows is inspired by the constructionmethods described in [13] and [48]. The construction is basedon representation of subspaces by Ferrers diagrams, optimalrank-metric codes, pending dots, and one-factorization of thecomplete graph. The definitions and results of the first sectionare taken from [13], [31], and [48].

A. Preliminaries for the Construction

1) Representation of Subspaces: For each rep-resented by the generator matrix in reduced row echelon form,denoted by , we associate a binary vector of lengthand weight , , called the identifying vector of , wherethe ones in are exactly in the positions where hasthe leading coefficients (the pivots). All the binary vectors oflength and weight can be considered as the identifying vec-tors of all the subspaces in . These vectors partition

into the different classes, where each class consistsof all subspaces in with the same identifying vector.The Ferrers tableaux form of a subspace , denoted by, is obtained from first by removing from each

row of the zeroes to the left of the leading coefficient;and after that removing the columns which contain the leadingcoefficients. All the remaining entries are shifted to the right.The Ferrers diagram of , denoted by , is obtained from

by replacing the entries of with dots. Given, the unique corresponding subspace can

be easily found.

Example 3: Let be the subspace in with the fol-lowing generator matrix in reduced row echelon form:

Its identifying vector is , and its Ferrerstableaux form and Ferrers diagram are given by

2) Lifted Ferrers Diagram Rank-Metric Codes: Let bea Ferrers diagram with dots in the rightmost column anddots in the top row. A code is an Ferrers diagramrank-metric code if all codewords of are matrices inwhich all entries not in are zeroes, it forms a rank-metric codewith dimension and minimum rank distance . The followingresult is the direct consequence from Theorem 1 in [13].


Lemma 12: Let , , , and let be an identi-fying vector, of length and weight three, in which the leftmostone appears in one of the first three entries. Let be the cor-responding Ferrers diagram and be a Ferrers diagramrank-metric code. Then, is at most the number of dots in ,which are not contained in its first row.A code which attains the bound of Lemma 12 will be called a

Ferrers diagram MRD code. A construction for such codes canbe found in [13].For a codeword , let denote the part

of related to the entries of in . Given a Ferrers diagramMRD code , a lifted Ferrers diagram MRD code is de-fined as follows:

This definition is the generalization of the definition of a liftedMRD code. The following lemma [13] is the generalization ofthe result given in Theorem 1.

Lemma 13: If is an Ferrers diagramrank-metric code, then its lifted code is anconstant dimension code.3) Multilevel Construction and Pending Dots: It was

proved in [13] that for any two subspaces ,we have , where de-notes the Hamming distance; and if , then

. These properties of thesubspace distance were used in [13] to present a multilevel con-struction, for a constant dimension code . In this construction,first a binary constant weight code of length , weight , andminimum Hamming distance is chosen. The codewords ofwill serve as the identifying vectors for . For each identifyingvector, a corresponding lifted Ferrers diagram MRD code withminimum rank distance is constructed. The union of theselifted Ferrers diagram MRD codes is an code.In the construction which follows, for , we also use a

multilevel method, i.e., we first choose a binary constant weightcode of length , weight , and minimum Hammingdistance . For each codeword in , a correspondinglifted Ferrers diagramMRD code is constructed. However, sincefor some pairs of identifying vectors the Hamming distance is 2,we need to use appropriate lifted Ferrers diagramMRD codes tomake sure that the final subspace distance of the code will be 4.For this purpose, we use a method based on pending dots in aFerrers diagram [48].The pending dots of a Ferrers diagram are the leftmost

dots in the first row of whose removal has no impact on thesize of the corresponding Ferrers diagram rank-metric code. Thefollowing lemma follows from [48].

Lemma 14 [48]: Let and be two subspaces inwith , such that the leftmost one of

is in the same position as the leftmost one of . Letand be the sets of pending dots of and , respectively.If and the entries in (of their Ferrerstableaux forms) are assigned with different values in at least oneposition, then .

Example 4: Let and be subspaces in which aregiven by the following generator matrices:

where , and the pending dots are emphasizedby circles. Their identifying vectors areand . Clearly, , while

.4) One-Factorization of Complete Graphs: A matching in a

graph is a set of pairwise disjoint edges of . A one-factoris a matching such that every vertex of occurs in exactly oneedge of the matching. A partition of the edge set in into one-factors is called a one-factorization. Let be a complete graphwith vertices. The following lemma is a well-known result[31, p. 476].

Lemma 15: has a one-factorization for all .A near-one-factor in is a matching with edges

that contain all but one vertex. A set of near-one-factors thatcontains each edge in precisely once is called a near-one-factorization. The following corollary is the direct conse-quence from Lemma 15.

Corollary 5: has a near-one-factorization for all .

Corollary 6: Let be a set of all binary vectors of lengthand weight 2.1) If is even, can be partitioned into classes, eachone has vectors with pairwise disjoint positions of ones;

2) If is odd, can be partitioned into classes, each onehas vectors with pairwise disjoint positions of ones.

B. First Construction

Construction I: Let and for odd(or for even ).1) Identifying Vectors: The identifying vector

corresponds to the lifted MRD code . Theother identifying vectors are of the form , where is oflength 3 and weight one, and is of length and weighttwo. We use all the vectors of weight two in the lastcoordinates of the identifying vectors. By Corollary 6, there is apartition of the set of vectors of length and weight 2 into

classes if is even (or into classes ifis odd), . We define

2) Ferrers Tableaux Forms and Pending Dots: All the Fer-rers diagrams which correspond to the identifying vectors fromhave one common pending dot in the first entry of the first


row. We assign the same value of in this entry of the Ferrerstableaux form for each vector in the same class. Two subspaceswith identifying vectors from different classes of have dif-ferent values in the entry of this pending dot. This is possiblesince the number of classes in is at most . On the remainingdots of Ferrers diagrams, we construct Ferrers diagram MRDcodes and lift them.Similarly, all the Ferrers diagrams, which correspond to the

identifying vectors from , have two common pending dots inthe first two entries of the first row. We assign the same valueof in these two entries in the Ferrers tableaux form for eachvector in the same class. Two subspaces with identifying vectorsfrom different classes of have different values in at least oneof these two entries. This is possible since the number of classesin is at most . On the remaining dots of Ferrers diagrams,we construct Ferrers diagram MRD codes and lift them.Finally, we lift Ferrers diagrams MRD codes which corre-

spond to the identifying vectors of .3) Code: Our code is a union of and the lifted codes

corresponding to the identifying vectors in , , and .

Example 5: For , there are different binary vec-tors of length and weight 2. We partition thesevectors into five disjoint classes ,

, ,, . The

identifying vectors of the code, besides , arepartitioned into three sets

To demonstrate the idea of the construction, we will only con-sider the set . The generator matrices in reduced row echelonform of the codewords with identifying vectors from are offour different types

where all the ’s are elements from . The suffixes (lastcoordinates) of the identifying vectors of the first two generatormatrices belong to , and of the last two matrices to . Allthese matrices have the same pending dot in the place of ,

. Then, we assign 0 in this place for the two firstmatrices and 1 in this place for the two last matrices

4) Analysis of the Construction:

Theorem 16: For satisfying , where

the code obtained in Construction I attains the bound of The-orem 10.

Proof: First, we prove that the minimum subspace distanceof is 4. Let , . We distinguish between threecases.• Case 1: If , then since theminimum distance of the is 4.

• Case 2: If and , then.

• Case 3: Assume .If , , , then clearly

.If , i.e., and have identifying vectors

, , where is of length 3, wedistinguish between two additional cases.— , . In this case,which implies .

— , , . If ,then . If , then byLemma 14, we have that .

Next, we calculate the size of . Recall that the identi-fying vectors are partitioned into classes. Note that since

, it follows that each one of the vectorsof weight 2 and length is taken as the suffix of someidentifying vector. Each such suffix (of length and weight2) is the identifying vector of a subspace in .By Lemma 12, each such subspace in is con-tained in exactly one codeword (since the first row of thegenerator matrix of the three-dimensional subspace is omittedby the lemma for the bound on ). The size of is

and the size of is . Hence, the

size of is . Theorem 10 implies that for

code , which contains an , wehave .


Remark 3: A code whose size attains the upperbound of Theorem 10was constructed in [13] and acode whose size attains this bound was constructed in [48].

C. Second Construction

For small alphabets, Construction I is modified as follows.Construction II: Let and for odd

(or for even ).The identifying vector corresponds to the

lifted MRD code . Let and

. For each other identifying vector, we partition thelast coordinates into or sets, where each one of thefirst sets consists of consecutive coordinates andthe last set (which exists if ) consists ofconsecutive coordinates. Since is always an eveninteger, it follows from Corollary 6 that there is a partition ofvectors of length and weight 2, corresponding to theset, , into classes .

We define,

, and, where denotes the zeroes

vector of length . Let

The identifying vectors (excluding ), of the code that weconstruct, are partitioned into the following three sets:

As in Construction I, we construct a lifted Ferrers diagramMRD code for each identifying vector, by using pending dots.Our code is a union of and the lifted codes corre-sponding to the identifying vectors in , , and .

Remark 4: The identifying vectors with two ones in the lastentries can also be used in Construction II, but their contributionto the final code is minor.In a similar way to the proof of Theorem 16, one can

prove the following theorem, based on the fact that thesize of the lifted Ferrers diagram MRD code obtained fromthe identifying vectors in , , is

.

Theorem 17: For satisfying , where

TABLE IIIRATIO BETWEEN AND THE BOUND IN (2)

Construction II generates an constant dimension

code with ,

which contains an .For all admissible values of , the ratio

, for the code generated byConstruction II, is greater than 0.988 for and 0.999 for

. Hence, the code almost attains the bound of Theorem 10.In the following table, we compare the size of codes obtained

by Constructions I and II (denoted by ) with the size of thelargest previously known codes (denoted by ) and with theupper bound (2) (for ).

The new ratio between the new best lower bound and theupper bound (2) with and is presented in Table III.One should compare it with Table II.

V. CONSTRUCTION FOR CODES

In this section, we introduce a construction ofcodes that attain the upper bound of Theorem 11, and are thelargest codes with these parameters. This construction is basedon 2-parallelism of subspaces in .A -spread in is a set of -dimensional subspaces

which partition (excluding the all-zero vector). We say thattwo subspaces are disjoint if they have only trivial intersection.A -spread in exists if and only if divides [37].Clearly, a -spread is a constant dimension code inwith the maximal possible minimum distance . A par-tition of all -dimensional subspaces of into disjoint-spreads is called a -parallelism. The following constructionis presented for .Construction III: Let be an obtained from an

. We will generate a new code that contains. The following new codewords (blocks) will form the

elements of .Let be a partition of all the subspaces of

into seven 2-spreads, each one of size 5, i.e., awell-known 2-parallelism in [3], [5], [54]. For each, , and each two subspaces ( canbe equal to ), we write and

, where , ,and . The two-dimensional subspace has fourcosets in . We construct the followingfour codewords in . The codewords are defined


TABLE IVPARTITION OF

by 15 nonzero vectors which are the nonzero vectors of afour-dimensional subspace as can be verified

(C.1)

(C.2)

(C.3)

(C.4)

In , there are two-dimensional subspaces,and hence there are 35 different choices for . Since the size ofa spread is 5, it follows that there are five different choices for. Thus, there are a total of codewords in

generated in this way. In addition to these 700 code-words, we add a codeword that contains all the points of .

Example 6: Apartition of into seven spreads is givenin Table IV, where each row corresponds to a spread.We illustrate the idea of Construction III by considering

one 2-spread and a coset of one element of the spread. Letbe a spread given by the first row of

the table, i.e., , ,, ,

. The four cosets of are given by

For the pair , , the following four subspaces , , ,and , belong to the code and correspond to the four types ofthe codewords, where corresponds to , , andfor every coset of we use a different color.

Theorem 18: Construction III generates anconstant dimension code that attains the bound of

Theorem 11 and contains an .Proof: First, we observe that the four types of codewords

given in the construction are indeed four-dimensional subspacesof . Each one of the codewords contains 15 different one-dimensional subspaces, and hence, each codeword contains 15different nonzero vectors of . It is easy to verify that all thesevectors are closed under addition in , thus each constructedcodeword is a four-dimensional subspace of .To prove that for each two codewords , we have

, we distinguish between three cases.• Case 1: . Since the minimum distance of

is 4, we have that .• Case 2: and . The code-words of form the blocks of an , ,and hencemeet each group in exactly one point. Each code-word of meets exactly three groups of . Hence,

for each and ,therefore, .

• Case 3: . If and have exactlythree points in common in (which correspond to atwo-dimensional subspace contained in ), then theyare disjoint in all the groups of . This is due to the fact thatthe points of in and the point of in correspondto either different cosets, or different blocks in the samespread. If and have exactly one point in common in

, then they have at most two points in common in atmost one group of . Thus, .

contains codewords. As explained in theconstruction, there are 701 codewords in . Thus, inthe constructed code , there are codewords.Thus, the code attains the bound of Theorem 11.

Remark 5: Construction III can be easily generalized for allprime powers , since there is a 2-parallelism infor all such , where is power of 2 [5]. Thus, from this con-struction, we can obtain a code with

, since the size of a 2-spread in is


and there are different cosets of a two-dimensionalsubspace in .In the following table, we compare the size of codes obtained

by Construction III and its generalizations for large (denotedby ) with the size of the largest previously known codes(denoted by ) and with the upper bound (2) (for and

).

Remark 6: In general, the existence of -parallelism inis an open problem. It is known that 2-parallelism

exists for and all [3], [54], and for each prime power, where is power of 2 [5]. There is also a 3-parallelism for

and [36]. Thus, we believe that Construction IIIcan be generalized to a larger family of parameters assumingthat there exists a corresponding parallelism.

VI. LINEAR CODES DERIVED FROM LIFTED MRD CODES

A lifted MRD code and the transversal design derived fromit can also be used to construct a linear code in the Hammingspace. In this section, we study the properties of such a linearcode, whose parity-check matrix is an incidence matrix of atransversal design derived from a lifted MRD code. Some ofthe results presented in this section generalize the results givenin [24]. In particular, the lower bounds on the minimum dis-tance and the bounds on the dimension of codes derived fromlifted MRD codes with coincide with the boundson low-density parity-check (LDPC) codes from partial geome-tries considered in [24]. Nevertheless, our goal in this section isto discuss the properties of the linear codes without taking intoaccount that some of them can be used as LDPC codes.For each codeword of an , we define its

binary incidence vector of length as follows:if and only if the point (one-dimensional subspace)is contained in .

Let be the binary matrix whose rows are theincidence vectors of the codewords of . By Theorem 6,this matrix is the incidence matrix of a ,with . Note that the rows of the incidencematrix correspond to the blocks of the transversal design,and the columns of correspond to the points of the transversaldesign. If in such a design (or, equivalently,for ), then is an incidence matrix of a net, the dualstructure to the transversal design [31, p. 243].An linear code is a linear subspace of dimension

of with minimum Hamming distance . Let be the linearcode with the parity-check matrix , and let be the linearcode with the parity-check matrix .The code has length and the code has

length . By Corollary 3, each column of hasones; since each -dimensional subspace contains

one-dimensional subspaces, each row has ones.

Remark 7: Note that if , then the column weight ofis one. Hence, the minimum distance of is 2. Moreover,consists only of the all-zero codeword. Thus, these codes are notinteresting, and hence, in the sequel, we assume that .

Lemma 19: The matrix obtained from ancode can be decomposed into blocks, where each block is a

permutation matrix.Proof: It follows from Lemma 5 that the related transversal

design is resolvable. In each parallel class, each element ofis contained in exactly one codeword of . Each class has

codewords, each group has points, and each code-word meets each group in exactly one point. This implies thatthe rows of related to each such class can be decom-posed into permutation matrices.

Example 7: A code and a code areobtained from the liftedMRD code . The in-cidence matrix for corresponding transversal design(see Example 2) is given by the following 16 12 matrix. Thefour rows above this matrix represent the column vectors for thepoints of the design.

Corollary 7: All the codewords of code , associated withthe parity-check matrix , and of code , associated with theparity-check matrix , have even weights.

Corollary 8: The minimum Hamming distance of andthe minimum Hamming distance of are upper boundedby .To obtain a lower bound on the minimum Hamming distance

of these codes, we need the following theorem known as theTanner bound [45].

Theorem 20: The minimum distance of a linear codedefined by an parity-check matrix with constant rowweight and constant column weight satisfies:

T1:

T2:

where is the second largest eigenvalue of .


To obtain a lower bound on and , we need to find thesecond largest eigenvalue of and , respectively.Note that since the set of eigenvalues of and the set ofeigenvalues of are the same, it is sufficient to find onlythe eigenvalues of .The following lemma is derived from [9, p. 563].

Lemma 21: Let be an incidence matrix for .The eigenvalues of are , , and with multi-plicities , and , respectively, where is a numberof blocks that are incident with a given point.By Corollary 3, in with

. Thus, from Lemma 21, we obtain the spec-trum of .

Corollary 9: The eigenvalues of are ,

, and 0 with multiplicities 1, , and

, respectively.Now, by Theorem 20 and Corollary 9, we have

Corollary 10:

Proof: By Corollary 9, the second largest eigenvalue ofis .We apply Theorem 20(T1) to obtain

By using Theorem 20, we also obtain lower bounds on

(4)

(5)

Note that the expression in (4) is negative for .For with and , the bound in (4)is larger than the bound in (5). Thus, we have , if

, and ; and ,otherwise.

We use the following result derived from [25, Th. 1] to im-prove the lower bound on .

Lemma 22: Let be an incidence matrix of blocks (rows)and points (columns) such that each block contains exactlypoints, and each pair of distinct blocks intersects in at mostpoints. If is a minimum distance of a code with the parity-check matrix , then

Corollary 11: .

Proof: By Lemma 22, with and ,since any two codewords in a lifted MRD code intersect in

at most -dimensional subspace, we have the followinglower bound on the minimum distance of :

Obviously, for all , this bound is larger or equal thanthe bound of Corollary 10, and thus, the result follows.

Let and be the dimensions of and ,respectively. To obtain the lower and upper bounds onand , we need the following basic results from linearalgebra [22]. For a matrix over a field , let denotethe rank of over .

Lemma 23: Let be a matrix, and let be the field ofreal numbers. Then1) .2) If and is a symmetric matrix with the eigen-value 0 of multiplicity , then .

Theorem 24:

Proof: First, we observe that, and .

Now, we obtain an upper bound on. Clearly, . By

Corollary 9, the multiplicity of an eigenvalue 0 ofis . Hence, by Lemma 23,

. Thus,

,

and .

Now, we obtain an upper bound on the dimension of the codesand for odd .

Theorem 25: Let be a power of an odd prime number.1) If is odd, then and

.

2) If is even, then and

.

Proof: We compute the lower bound on toobtain the upper bound on the dimension of the codes and. First, we observe that . By

[8], the rank over of an integral diagonalizable square ma-trix is lower bounded by the sum of the multiplicities of theeigenvalues of that do not vanish modulo 2. We considernow . By Corollary 9, the second eigenvalue of

is always odd for odd . If is odd, then the firsteigenvalue of is also odd. Hence, we sum the multiplic-ities of the first two eigenvalues to obtain

. If is even, then the first eigenvalue


is even, and hence, we take only the multiplicity of the secondeigenvalue to obtain . Theresult follows now from the fact that the dimension of a code isequal to the difference between its length and .

Remark 8: For even values of , the method used inthe proof for Theorem 25 leads to a trivial result, sincein this case, all the eigenvalues of are even andthus, by [8], we have . But clearly, byLemma 19, we have . Thus, for even ,

and.

Note that for odd and odd , the lower and the upperbounds on the dimension of and are the same. Therefore,we have the following corollary.

Corollary 12: For odd and odd , the dimensionsand of the codes and , respec-

tively, satisfy , and

.Finally, and can also be viewed as LDPC codes ob-

tained from designs [2], [23]–[25], [28]–[30], [46], [50], [51],[55]. Some preliminary results in this direction can be foundin [38] and [40]. The performance of LDPC codes based ontransversal designs, in an additive white Gaussian noise channelusing sum–product decoding algorithm, was studied in [24].The codes presented in [24] correspond to our code , where

. It was shown [24] that the codes with columnweight three have a significant improvement in their decodingperformance over random codes with the same length and rate.Moreover, when compared to the codes of the same length basedon finite geometries [28], the codes from transversal designshave a higher rate and a lower decoding complexity at largersignal-to-noise ratios [24]. Finally, only in this case , thegirth of the corresponding graph is 6, while in the other cases,the girth is 4, which is generally an unwanted property for LDPCcodes.

VII. CONCLUSION AND FUTURE RESEARCH

Lifted MRD codes are considered. Properties of these codes,especially when viewed as transversal designs are proved.Based on this design, new upper bounds and constructions forconstant dimension codes which contain lifted MRD codesas subcodes are given. The incidence matrix of the design(which represents also the codewords of the lifted MRD code)is considered as a parity-check matrix of a linear code in theHamming space. Properties of these linear codes are proved.We conclude with a list of open problems for future research.1) What are the general upper bounds on a size of an

code which contains a lifted MRD code?2) Are the upper bounds of Theorems 10 and 11 and relatedbounds for other parameters attained for all parameters?

3) Can the codes constructed in Constructions I, II, and IIIbe used, in a recursive method, to obtain new bounds on

for larger ?4) One of the main research problems is to improve the lowerbounds on , with codes which do not contain

the lifted MRD codes. Only such codes can close the gapbetween the lower and the upper bounds onfor small and small (e.g., the seven codes formentioned in Section I).

5) We did not check the linear codes obtained from liftedMRD codes as LDPC codes. It is intriguing to find whichproperties have LDPC codes obtained from lifted MRDcodes? The bounds given in Section VI can be of help inthis direction. In addition, we would like to know the per-formance of these codes with various decoding algorithms[11], [34].

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Tuvi Etzion (M’89–SM’94–F’04) was born in Tel Aviv, Israel, in 1956. Hereceived the B.A., M.Sc., and D.Sc. degrees from the Technion—Israel Instituteof Technology, Haifa, Israel, in 1980, 1982, and 1984, respectively.From 1984 he held a position in the department of Computer Science at

the Technion, where he has a Professor position. During the years 1986–1987he was Visiting Research Professor with the Department of Electrical En-gineering—Systems at the University of Southern California, Los Angeles.During the summers of 1990 and 1991 he was visiting Bellcore in Morristown,New Jersey. During the years 1994–1996 he was a Visiting Research Fellowin the Computer Science Department at Royal Holloway College, Egham,England. He also had several visits to the Coordinated Science Laboratoryat University of Illinois in Urbana-Champaign during the years 1995–1998,two visits to HP Bristol during the summers of 1996, 2000, a few visits to thedepartment of Electrical Engineering, University of California at San Diegoduring the years 2000–2012, and several visits to the Mathematics departmentat Royal Holloway College, Egham, England, during the years 2007–2009.His research interests include applications of discrete mathematics to prob-

lems in computer science and information theory, coding theory, and combina-torial designs.Dr Etzion was an Associate Editor for Coding Theory for the IEEE

TRANSACTIONS ON INFORMATION THEORY from 2006 till 2009.

Natalia Silberstein was born in Novosibirsk, Russia, in 1977. She received theB.A. degree in Computer Science and Mathematics, M.Sc. degree in AppliedMathematics, and Ph.D. degree in Computer Science from the Technion—Is-rael Institute of Technology, Haifa, Israel, in 2004, 2007, and 2011, respectively.Since November 2011, she is a postdoctoral fellow in the Wireless Networkingand Communications Group, at the Department of Electrical and Computer En-gineering at the University of Texas at Austin, USA. Her research interests in-clude coding theory, combinatorial designs, and their application to distributedstorage systems and network coding.

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