IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003 2253
[5] P. Camion, Abelian codes, Univ. Wisconsin, Madison, Math. Res. Ctr.,Tech. Rep 1059, 1971.
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Cubic Self-Dual Binary Codes
Alexis Bonnecaze, Member, IEEE, Anne Desideri Bracco,Steven T. Dougherty, Luz R. Nochefranca, and
Patrick Sol, Member, IEEE
AbstractWe study binary self-dual codes with a fixed point free auto-morphism of order three. All binary codes of that type can be obtained by acubic construction that generalizes Turyns. We regard such cubic codesof length 3 as codes of length over the ring . Classical notions ofType II codes, shadow codes, and weight enumerators are adapted to thatring. Two infinite families of cubic codes are introduced. New extremal bi-nary codes in lengths 66 are constructed by a randomized algorithm.Necessary conditions for the existence of a cubic [72 36 16]Type II codeare derived.
Index TermsAutomorphism group, codes over rings, self-dual codes.
I. INTRODUCTION
The construction of binary self-dual codes with an automorphism ofgiven odd order has received a lot of attention over the years [14].
In this correspondence, we consider the case of an automorphismof order three without a fixed point. It was shown in [15] that all suchcodes can be obtained by a generalized cubic construction from a binarycode and a quaternary code both of length `. From now on, we will callsuch codes cubic.
We view cubic codes as codes of length ` over the ring2
4
. Westudy self-dual codes over that alphabet and adapt to that ring the clas-sical tools in the study of self-dual codes: Type II codes, shadow codes,weight enumerators, and invariant theory. We give two infinite familiesof cubic self-dual codes related to quadratic residue (QR) codes andReedMuller (RM) codes, respectively. We give examples of extremalself-dual cubic codes for ` 22, and thereby examples of the applica-tion of the tools developed. Necessary conditions for the existence of aputative cubic Type II [72; 36; 16] are derived.
Manuscript received October 3, 2001; revised April 8, 2003. This work wasperformed while A. Bonnecaze was visiting INRIA project GALAAD at SophiaAntipolis, France.
A. Bonnecaze is with the IAAI, 13003 Marseille, France (e-mail: [email protected]).
A. Desideri Bracco and P. Sol are with the CNRS, I3S ESSI, 06 903 SophiaAntipolis, France (e-mail: [email protected]; [email protected]).
S. T. Dougherty is with the Department of Mathematics, University ofScranton, Scranton, PA 18510 USA (e-mail: [email protected]).
L. R. Nochefranca is with the Department of Mathematics, University ofthe Philippines, Diliman, 1101 Quezon City, Philippines (e-mail: [email protected]).
Communicated by S. Litsyn, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2003.815800
0018-9448/03$17.00 2003 IEEE
2254 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003
II. NOTATION AND DEFINITIONS
We want to study binary codes formed by the cubic type constructionof [15]. Binary codes C of length 3` are formed by this constructionfrom a binary code C2
of length ` and a quaternary code C4
of length`. More specifically, ifA,B, andX are binary vectors of length ` thenlet
U =X +A
V =X +B
W =X +A +B
and, writing4
=
2
(!)
(X; A+ !B) := (U jV jW )
defines a map from `2
`
4
!
3`
2
. With these notations, let C =(C
2
; C
4
).
In the special case where C4
admits a binary basis we recover the(a + xjb + xja + b + x) construction of [9, p. 587] that yieds, in aparticular case, Turyns construction of the Golay code [9, p. 588].
By analogy with the theory of4
-codes we shall call this map a Graymap. Further, by Lee weight and Lee distance over2
4
, we shallmean the standard Hamming weight and Hamming distance of the Grayimage.
Alternatively we think of the vector (U jV jW ) as a vector of length` over2
4
with the jth coordinate as (Xj
; A
j
+ !B
j
).
For a code C over2
4
we define the symmetric weight enu-merator as
swe
C
(a; b; c; d) =
c2C
a
n (c)
b
n (c)
c
n (c)
d
n (c) (1)
where ni
is the number of triples (uj
; v
j
; w
j
) with Hamming weighti = 0; 1; 2; 3 for 1 j `.
It can be shown that the Gray image (C2
; C
4
), where C2
is a bi-nary code andC4
is a quaternary code, is self-dual (resp., Type II) if andonly if C2
is self-dual (resp., Type II) and C4
is Hermitian self-dual,see [15]. A self-dual code is said to be Type II if its Gray image is TypeII and Type I otherwise.
Note, however, that if C2
and C02
are isomorphic binary codes andC
4
and C 04
are isomorphic quaternary codes, this does not necessarilymean that (C2
; C
4
) and (C 02
; C
0
4
) are isomorphic. This innocentobservation will be used in the examples section to construct extremalbinary codes by a randomized algorithm.
For any vector v, the Hamming weight wH
(v) is the number ofnonzero coordinates of the code. We define the Hamming weight enu-merator by
W
C
(x; y) =
c2C
x
nw (c)
y
w (c)
: (2)
Often, the x is replaced by 1.For two vectors v; w in (2
4
)
` we define the inner product by
[v; w] =
`1
i=0
(v
i
; w
i
) (3)
where
(v
i
; v
0
i
) = xx
0
+Tr ((a+ !b)(a
0
+ !b
0
)) (4)
with vi
= (x; (a+!b)) and with v0i
= (x
0
; (a
0
+!b
0
)) andTr denotesthe trace from4
down to2
.
Lemma 2.1: The inner product given in (3) is equivalent to the innerproduct of the Gray image.
Proof: The binary inner product of the Gray image is
(X + A) (X
0
+A
0
) + (X +B) (X
0
+B
0
)
+(X +A+B) (X
0
+A
0
+B
0
) = X X +A B
0
+B A
0
and the inner product in (3)
=X X
0
+Tr (A A
0
+ A B
0
+B B
0
+ !(A B
0
+B A
0
))
=X X
0
+A B
0
+B A
0
giving the result.
III. FIRST PROPERTIES
Proposition 3.1: Let C = (C2
; C
4
), then
W
C
(x; y) = swe
C
(x; 0; y; 0) =
1
jC
2
j
swe
C
(x; y; y; x) (5)andW
C
(x; y) = swe
C
(x; 0; 0; y) =
1
jC
4
j
swe
C
(x; y; x; y): (6)
Proof: Write4
=
2
(!). Note that
2w
H
(A+ !B) = w
H
(A) + w
H
(B) + w
H
(A+B):
Counting in two ways, we find that
w
H
(U + V ) + w
H
(U +W ) + w
H
(V +W ) = 2n
1
+ 2n
2
:
Moreover, by definition of U; V; andW we have
w
H
(A) = w
H
(U + V )
w
H
(B) = w
H
(V +W )
w
H
(A+B) = w
H
(U + V )
and the evaluation ofWC
follows.To evaluateWC
observe that wH
(U + V +W ) = n
1
+ n
3
. Theresult follows.
Lemma 3.2: If C = (C2
; C
4
) then
W
C
(x; y) = swe
(C ;C )
(x
3
; x
2
y; xy
2
; y
3
)
In the proof of the following theorem, we shall assume some famil-iarity with [22]. Note that a code over a Cartesian product of alphabetsis equivalent to a Cartesian product of codes over the said alphabets.
Theorem 3.3: If C = C2
C
4
with C2
binary and C4
quaternaryboth linear then
swe
C
(a; b; c; d) =
1
jCj
T swe
C
(a; b; c; d) (7)
where
T =
1 3 3 1
1 1 1 1
1 1 1 1
1 3 3 1
acts on sweC
(a; b; c; d) by linear substitution in the variablesa; b; c; d.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003 2255
Proof: Since the ring2
4
is quasi-Frobenius with admis-sible character ((x; y)) := (1)x+Tr (y), then by [22], there is alinear transformation T satisfying the condition of the theorem. Sincethe Gray image of a self-dual code is self-dual, the explicit expansionfor T follows by Lemma 3.2.
IV. CONSTRUCTIONS OF CUBIC CODES
A. Cyclic and Circulant CodesIn this section, we construct two infinite families of Type I cubic
codes: subtracted QR and subtracted RM. We take for granted the fol-lowing fact from [15]: a code C is cubic iff its automorphism groupAut (C) contains a fixed-point-free (f.p.f.) element of order 3.
Proposition 4.1: If a binary code C is cyclic of length 3` or, moregenerally `-quasi-cyclic of length 3`m (m 1 integral) then C iscubic.
Proof: If Aut (C) contains an f.p.f. element, say, of order 3mthen := m is also f.p.f. and satisfies 3 = 1.
There is an extremal cyclic Type I code of length 42 [3]. It is there-fore cubic.
B. Quadratic Residue CodesLet QR (p) denote the QR code of length p + 1.Theorem 4.2: If p is a prime 23 (mod 24) then QR (p) is a
cubic self-dual Type II code of length p + 1.Proof: It is well known [9, Ch. 16, Lemma 14] that PSL(2; p)
contains a permutation , say, consisting of two disjoint cycles oflength p+12
. Let be raised to the power p+16
. We see that isf. p. f. of order 3.
The Golay code is the case p = 23 of Theorem 4.2. This is equivalentto Turyns construction. The case p = 47 is described in [12]. Bya subtracted QR code (denoted here by SQR (p)) we shall mean theself-dual code obtained by subtraction at the places 0 and 1. Moreformally
SQR (p) := fs 2 p12
j(0; s; 0) 2 QR (p) _ (1; s; 1) 2 QR (p)g:
Theorem 4.3: If p is a prime 1 (mod 8) then SQR (p) is a2-quasi-cyclic self-dual Type I code of length p 1. More generally,its automorphism group contains a dihedral group of order p 1. If,furthermore, p 1 (mod 3), then SQR (p) is cubic.
Proof: It is straightforward to check from the definition thatSQR (p) is self-dual. Since p 1 is not a multiple of 8 we see thatSQR (p) is a Type I code. We proceed to show that it is left whollyinvariant by an f. p. f. permutation of order p12
. By finite-field theory,we know that p
contains a cubic root , say, of unity. Now the matrix
A :=
0
0
1
belongs to SL(2; p), and satisfies A = I . Because A is diag-onal, its image A in PSL(2; p) fixes 0 and1. Now SQR (p) is alsoleft invariant by the involution B of antecedent in SL(2; p)
B :=
0 1
1 0
:
It can be checked that hA; Bi is a dihedral group of order p 1. Thelast assertion follows by Proposition 4.1.
We conjecture that the automorphism group of SQR (p) is thedihedral group Dp1
. By Dirichlets theorem on primes in arith-metic progression there are infinitely many primes satisfying thehypothesis of Theorem 4.3. The first few relevant values of p arep = 31; 79; 103; . . . For p = 31, the Type I code SQR (31) isextremal of parameters [30; 15; 6].
C. ReedMuller (RM) CodesLet RM (r; m) denote the RM code of order r and length 2m. It is
well known [9, Theorem 4, p. 375] that this code is self-dual for modd and r = m12
. Note that, furthermore, its coordinate places areindexed by v 2 m2
[9, Fig. 13.4, p. 376]. Let SRM (m) denote theself-dual code obtained by subtraction at the places v = 0 and v = 1(the all-one vector) from the preceding code.
Theorem 4.4: If m 3 is odd then SRM (m) is a pure doublecirculant self-dual Type I code of length 2m 2.
Proof: First, we determine the automorphism group ofSRM (m). Recall that the automorphism group of the RM (r; m)shortened at the position v = 0 is GL(m; 2) acting on m2
n 0.
Assume that 1 is a fixed point of A 2 GL(m; 2), which means, interms of linear algebra, that it is an eigenvector ofA for the eigenvalue1. Let V denote an orthogonal complement of the line spanned by1 in m2
. Then V is stable by action of A, and any A satisfyingA1 = 1 is uniquely determined by its restriction to V . We thus seethat the stabilizer of f0; 1g in the affine group Aut (RM (r; m)) isisomorphic to GL(m 1; 2).
Next, we show that Aut (SRM (m)) contains a f, p. f. element oforder 2m1 1. In other words, we need an element of GL(m 1; 2) which does not fix any point of V but 0. By finite-field theory,
2
contains an element of order 2m1 1. Identifying m12
and2
, we see that the matrix of the endomorphism x 7! xsatisfies our requirements.
For m = 5, the Type I code SRM (5) is extremal of parameters[30; 15; 6].
V. SHADOWS
For a binary codeC , the shadow is well defined (see [3]). We extendthis definition to codes over2
4
. Let C be a Type I code over2
4
. Let D0
be the subcode consisting of those vectors whoseimage under has weight congruent to 0 (mod 4). This subcode isof index 2 inC . LetD2
= CD
0
andD?0
= C[S, with the shadowS = D
1
[D
3
. The glue group forD?0
=D
0
is the Klein group of order4 for all `.
Theorem 5.1: If C is a self-dual code over2
4
then
swe
D
(a; b; c; d) =
1
2
(swe
C
(a; b; c; d) + swe
C
(a; ib; c; id))
(8)andswe
S
(a; b; c; d) =M swe
C
(a; ib; c; id): (9)
Proof: The weight enumerator sweC
(a; ib; c; id) negatesonly those vectors whose Gray image has singly even weight. The stan-dard computation of [3] gives the rest.
A. Shadow SumLet C be a self-dual code of length ` over2
4
and let Di
bedefined as above. Note that, in the notation of [3],(Di
) = (C)
i
fori = 0; 2 and up to labeling for i = 1; 3. We note that the Gray imageof the shadow is the shadow of the Gray image. Then, since the inner
2256 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003
TABLE IORTHOGONALITY RELATIONS FOR 2 (mod 4)
TABLE IIORTHOGONALITY RELATIONS FOR 0 (mod 4)
product defined on the ambient space is equivalent to the inner productdefined by the Gray images by Lemma 2.1, we obtain the orthogonalityrelations given in Tables I and II.
We shall now describe the shadow sum construction for codes over2
4
using the notation in [7].Let C and C0 be self-dual codes of length ` and `0, respectively.
Define Di
and D0i
as above.Let
E = (D
0
; D
0
0
) [ (D
1
; D
0
1
) [ (D
2
; D
0
2
) [ (D
3
; D
0
3
) (10)andF =(D
0
; D
0
0
) [ (D
1
; D
0
3
) [ (D
2
; D
0
2
) [ (D
3
; D
0
1
) (11)
where
(D
i
; D
0
j
) = f(v; w) j v 2 D
i
; w 2 D
0
j
g:
The codesE and F are called the shadow sum of C andC 0 dependingon which case produces a self-dual code and is denoted by C s
C
0
.
Theorem 5.2: Let C and C 0 be self-dual codes over2
4
oflength ` and `0, respectively. If ` `0 0 (mod 4), then E is aself-dual code of length ` + `0. If ` `0 2 (mod 4), then F is aself-dual code of length ` + `0. Moreover, E and F are Type II codesif and only if ` + `0 0 (mod 8).
Proof: Linearity follows from the fact that the glue group is theKlein group of order 4 for both codes C and C 0. The fact that they areself-orthogonal follows from Tables I and II. Self-duality follows byconsidering their cardinality, i.e.,
jEj = jF j = 4jD
i
jjD
0
j
j = 4
jCj
2
jC
0
j
2
= jCkC
0
j = 8 :
as desired.For Type II, the weights in the shadow are `2
(mod 4) and `2
(mod 4), respectively. So in (Di
; D
0
j
), i = 1; 3 the weights are`+`
2
(mod 4). The weights in (D0
; D
0
0
) and (D2
; D
0
2
) are always0 (mod 4).
Corollary 5.3: If E and F are the shadow sum of C and C 0 asdefined above then
swe
E
(a; b; c; d)
=
3
i=0
swe
D
(a; b; c; d) swe
D
(a; b; c; d)
swe
F
(a; b; c; d)
=
i2f0; 2g
swe
D
(a; b; c; d) swe
D
(a; b; c; d)
+
i2f1; 3g
swe
D
(a; b; c; d) swe
D
(a; b; c; d):
(12)
Lemma 5.4: If A; B 2 n2
then
2w
H
(A+ !B) = w
H
(A) + w
H
(B) + w
H
(A+B):
In particular, d(C4
)
d(C)
2
.
Proof: Observe that wH
(A + !B) = w
H
(A _ B), where _stands for the logical, or inclusive OR. By inspection of the Karnaughtable for the Boolean functionsAi
; B
i
; A
i
_B
i
, andAi
+B
i
for eachcoordinate i the result follows.
Lemma 5.5: If x 2 C2
a Type II binary code and a + !b 2 C4
a
quaternary self-dual code then wH
((x; a+!b))w
H
(x)(mod 4).
Proof: Since (C2
; C
4
) is Type II we see that
w
H
((X; A+!B))w
H
((X; 0))+w
H
((0; A+!B))(mod 4):
NowwH
((0; A+!B)) = 2w
H
(A+!B) by Lemma 5.4. SinceC4
is Type IV we see that wH
(A + !B) is even. Since
w
H
((X; 0)) = 3w
H
(X) w
H
(X) (mod 4)
the result follows.
Theorem 5.6: If C2
C
4
is a code C over2
4
then Di
=
(C
2
)
i
C
4
where (C2
)
0
is the subcode of doubly-even vectors of thebinary code C2
and then (C2
)
i
are defined as usual.Proof: The fact thatD0
= (C
2
)
0
C
4
follows from Lemma 5.5and the rest from a straightforward calculation.
Theorem 5.7: If C2
C
4
is a code over2
4
and C 02
C
0
4
is acode over2
4
then
C
s
C
0
=(C
2
C
0
2
) (C
4
C
0
4
) (13)and(C
2
; C
4
)
s
(C
0
2
; C
0
4
) =(C
2
s
C
0
2
; C
4
C
0
4
): (14)
Proof: Follows from Lemma 5.5.The following corollary will be used in constructing examples.
Corollary 5.8: If C and C 0 are cubic binary codes so is C C 0.
VI. INVARIANTS
The swe of a Type II code is held invariant by the matrix
1
=
1 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003 2257
since the weights in the Gray image are congruent to 0 (mod 4), by
2
=
1 0 0 0
0 i 0 0
0 0 1 0
0 0 0 i
since C2
is Type II binary (see Proposition 3.1), and by
2
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
since C4
is a Type IV quaternary code (see Proposition 3.1), and byM =
1
p
8
T since it is self-dual.Let GII
= h
1
;
2
;
3
; Mi. A Magma computation gives thatjG
II
j = 1152 and the Molien series is
1 + 3t
8
+ 9t
16
+ 26t
24
+ 52t
32
+ 91t
40
+ 155t
48
+ 237t
56
+341t
64
+ 484t
72
+ 654t
80
+ 855t
88
+ 1109t
96
+ :
The swe of a Type I code is held invariant by 21
since the weightsin the Gray image are congruent to 0 (mod 2), and by
3
andM . LetG
I
= h
2
1
;
3
; Mi. By a Magma computation we have that jGI
j =
96 and the Molien series is
1 + t
2
+ 2t
4
+ 3t
6
+ 6t
8
+ 8t
10
+ 13t
12
+ 16t
14
+ 24t
16
+ 32t
18
+ 42t
20
+ 50t
22
+ 68t
24
+ 82t
26
+ 100t
28
+ 118t
30
+ 145t
32
+ 168t
34
+ 200t
36
+ 227t
38
+ 266t
40
+ 305t
42
+ 349t
44
+ 388t
46
+ 447t
48
+ 499t
50
+ 558t
52
+ 617t
54
+ 692t
56
+ 760t
58
+ 843t
60
+ 918t
62
+ 1012t
64
+ 1106t
66
+ 1208t
68
+ 1302t
70
+ 1426t
72
+ 1540t
74
+ 1664t
76
+ 1788t
78
+ 1935t
80
+ 2072t
82
+ 2230t
84
+ 2377t
86
+ 2550t
88
+ 2723t
90
+ 2907t
92
+ 3080t
94
+ 3293t
96
+ 3493t
98
+ :
VII. CUBIC CODES OF LENGTH 72
It might help to keep in mind the following easy but useful proposi-tion.
Proposition 7.1: If C = (C2
; C
4
) then
d(C) min(3d(C
2
); 2d(C
4
)):
Proof: Observe that ifX 2 C2
thenC contains a word of weight3w
H
(X). Similarly, ifA+!B 2 C4
thenC contains a word of weight2w
H
(A+ !B) = w
H
(A) + w
H
(B) + w
H
(A+B).
1) ` = 2Let i2
denote the binary repetition code of length 2. Theunique binary self-dual code of length 6 is obtained byi
6
= (i
2
; i
2
4
).
2) ` = 4The shadow sum of i6
with itself yields a cubic code of length12 with weight enumerator:
1 + 15y
4
+ 32y
6
+ 15y
8
+ y
12
:
3) ` = 6By [12], we know that the (extremal) extended QR code iscubic. The minimum Lee distance of a self-dual code of length
6 over2
4
is 4. It is a simple calculation to see from theinvariants of degree 6 for GI
that its Gray image must be
1 + 9y
4
+ 75y
6
+ 171y
8
+ 171y
10
+ 75y
12
+ 9y
14
+ y
18
:
It is shown in [3] that there are two possible weight enu-merators for Type I codes over length 18 with minimum dis-tance 4. The code (d10
e
7
f
1
)
+ has weight enumerator 1 +17y
4
+ and hence cannot be 6-quasi-cyclic. The code d3+6
is (C2
; C
4
) where C4
is the hexacode with weight enumer-ator 1 + 45y4 + 18y6 and C2
= I
3
2
with weight enumerator1 + 3y
2
+ 3y
4
+ y
6
. Notice that no information about theautomorphism group of (d10
e
7
f
1
)
+ is needed to show that itcannot be 6-quasi-cyclic.
4) ` = 8The highest minimum Lee distance of a self-dual code oflength 8 over2
4
is 8. Hence, its Gray image must bethe Golay code with weight enumerator
1 + 759y
8
+ 2576y
12
+ 759y
16
+ y
24
:
Moreover, the binary code must have weight enumerator 1 +14y
4
+ y
8
, that is. it is the [8; 4; 4] Hamming code.
5) ` = 10Using invariant theory, we can see that the minimum Lee dis-tance of a self-dual code of length 10 over2
4
is at most8. But since the highest minimal distance of a [10; 5] binaryself-dual code is 2 (see [3]), the best minimal distance we canreach is 6. It is attained for the inverse Gray image of SQR (31)and SRM (5). These codes are extremal.
6) ` = 12The minimum Lee distance of a self-dual code of length 12over2
4
is at most 10. However, the highest minimumdistance for a self-dual code of length 36 is just 8. Two suchcodes appear in [3] as d3 andR2. But neither is cubic as can beseen by electronic inspection of their permutation groups. Weconstruct a cubic code C36
= (C
2
; C
4
) with the parame-ters [36; 16; 8], and an automorphism group of order 288. Wetook C2
= d
+
12
and C4
= (C
2
4
) with the permutation = (2; 9; 8; 5)(3; 12; 4; 7)(6; 11; 10). That permutationwas found by using the function Random of Magma. Usingthe notation of Harada in [6], this code has W36; 1
as weightenumerator but it is not equivalent to any of the ten codesconstructed by Harada, their automorphism groups being dif-ferent.
7) ` = 14Taking C2
= D
14
[19]
= (1; 13; 10)(2; 3; 5; 12; 9; 7; 8; 6)(4; 11; 14)
andC4
an extended QR code leads to an extremal [42; 21; 8].There are 30 such codes according to [4]. The one we foundis not cyclic unlike the one given in [3]. Its weight enumeratorisW1
; = 0 in the sense of [4]. Its automorphism group iscyclic of order 3.
8) ` = 16Taking C2
= F
16
[19]
= (1; 6; 4)(2; 11; 7; 9; 13; 5; 12)(3; 15; 10)(14; 16)
and C4
= R
16
a code of [16, p. 313], we find an extremal[48; 24; 10]. Note that the automorphism group of the code
2258 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003
(N
1
) of [3] which shares the same weight enumerator istrivial.
9) ` = 18TakingC2
= H
18
; I
18
, respectively, [19] andC4
= S
18
[13],we find, in the notations of [20], a [54; 27; 10] with weightenumeratorW1; = 0, and, also a [54; 27; 10] with weightenumeratorW2; = 12. The latter has automorphism groupof size three and is, therefore, different from those constructedin [20], whose automorphism group contains a 7-element andfrom the one constructed by Tsai [21], whose automorphismgroup is trivial.
10) ` = 20Taking C2
= J
20
, and the two extremal possible C4
[13],we find (for three different s) three nonequivalent extremal[60; 30; 12] codes with weight enumerators (like the 5 in [5])
1 + 3195y
12
+
and automorphism groups of size 3; 6; 12; respectively.
11) ` = 22Taking C2
, the unique extremal binary code in length 22 andC
4
a quaternary Hermitian self-dual code found in [11], wefound two [66; 33; 12] within the notation of [4]weightenumeratorW1
and = 21; 30; respectively. Our two codesare therefore different from the codes constructed in [4, Sec.3.1].
12) ` = 24A problem of old standing in coding theory is the existenceof an extremal Type II self-dual binary code in length 72 [8].If such a code is of the form (C2
; C
4
), then, by [15, Corol-lary 7.2], we know that C2
is the extended Golay code, andthatC4
is a Hermitian self-dual Type IV code with parameters[24; 12; 8]. According to [18], there are at least 205 nonequiv-alent such codes. However, most of the codes in [18] are noteligible for being C4
s, as the next result shows.
Theorem 7.2: If (C2
; C
4
) is a [72; 36; 16] Type II code, then C4
cannot be invariant under an f. p. f. permutation of order three.
We prepare for the proof by deriving first a pair of lemmas.The first lemma is interesting in its own right.
Lemma 7.3: A [72; 36; 16] Type II code cannot be invariant underan f. p. f. permutation of order nine.
Proof: By [15, Theorem 5.1] such a code could be written as
C := f(c
0
; . . . ; c
8
)jc
i
= x+Tr (y
3i
) + Tr (z
i
) x 2 N
1
; y 2 N
2
; z 2 N
6
g
whereNi
is an [8; 4] code over2
. Now, by the nonic analog of Propo-sition 7.1, the codeN6
should be an [8; 4; 8] code, a fact which violatesthe Singleton bound.
The second Lemma is more technical.
Lemma 7.4: If C2
is cubic and C4
is invariant under an f. p. f. per-mutation of order three then (C2
; C
4
) is invariant under an f. p. f.permutation of order nine.
Proof: (sketch) Write X = (x + a; x + b; x + a + b) for atypical codeword of C2
, with x 2 c2
, a binary code and a + !b 2 c4
an4
-code. Similarly, by the van der Monde construction of [15], wecan write an arbitrary codeword of C4
as
A+ !B = (F
0
+ F
1
+ F
2
; F
0
+ !F
1
+ !
2
F
2
; F
0
+ !
2
F
1
+ !F
2
)
withFi
2 q
i
a quaternary code. Substituting into the cubic construction
(X +A; X +B; X +A +B)
for (C2
; C
4
), we see that this code admits a nonic constructionfrom the codes c2
; c
4
; q
0
q
1
q
2
, which is consistent with theChinese Remainder Theorem decomposition [15, eq. (3)] for the ring2
[Y ]=(Y
9
1). Indeed, the decomposition of Y 9 1 in irreduciblefactors is
Y
9
1 = (Y 1)(Y
2
+ Y + 1)(Y
6
+ Y
3
+ 1):
We are now ready for the proof of Theorem 7.2.
Proof: By the above discussion C2
is the extended Golay code,which, by Turyns construction, is cubic. By Lemma 7.4 and the hy-pothesis made on C4
, the code (C2
; C
4
) is then invariant under anf. p. f. permutation of order nine, a fact which contradicts Lemma 7.3.
ACKNOWLEDGMENT
The authors wish to thank the referees for their constructive criti-cism, and A. Otmani for sending them the codeC4
for ` = 22. Magmacomputations were performed on the machines of UMS Medicis
REFERENCES[1] E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, Type II codes,
even unimodular lattices and invariant rings, IEEE Trans. Inform.Theory, vol. 45, pp. 11941205, May 1999.
[2] R. A. Brualdi and V. Pless, Weight enumerators of self-dual codes,IEEE Trans. Inform. Theory, vol. 37, pp. 12221225, July 1991.
[3] J. H. Conway and N. J. A. Sloane, A new upper bound on the minimumdistance of self-dual codes, IEEE Trans. Inform. Theory, vol. 36, pp.13191333, Nov. 1990.
[4] S. T. Doughterty, T.A. Gulliver, and M. Harada, Extremal binaryself-dual codes, IEEE Trans. Inform. Theory, vol. 43, pp. 20362046,Nov. 1997.
[5] T. A. Gulliver and M. Harada, Weight Enumerators of extremal singlyeven [60,30,12] codes, IEEE Trans. Inform. Theory, vol. 42, pp.658659, Mar. 1996.
[6] M. Harada, New extremal self-dual codes of lengths 36 and 38, IEEETrans. Inform. Theory, vol. 45, pp. 25412543, Nov. 1999.
[7] S. T. Dougherty and P. Sol, Shadows of codes and lattices, inProc. Asian Math Conf. 2000, T. Sunada, P. W. Sy, and Y. Lo,Eds. Singapore: World Scientific, 2002, pp. 139152.
[8] Length 72 problem. [Online]. Available: http://academic.uofs.edu/fac-ulty/Doughertys1/72.htm.
[9] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error CorrectingCodes. Amsterdam, The Netherlands: Elsevier/ North-Holland, 1977.
[10] http://www.unilim.fr/pagesperso/philippe.gaborit/. [Online][11] P. Gaborit and A. Otmani, Experimental constructions of self-dual
codes, available from [10].[12] W. C. Huffman, Automorphisms of codes with applications to extremal
doubly even codes of length 48, IEEE Trans. Inform. Theory, vol.IT-28, pp. 511521, May 1982.
[13] , Characterization of quaternary extremal codes of lengths 18 and20, IEEE Trans. Inform. Theory, vol. 43, pp. 16131616, Sept. 1997.
[14] , Codes and groups, in Handbook of Coding Theory, V. S. Plessand W. C. Huffman, Eds. Amsterdam, The Netherlands: North-Hol-land, 1998, vol. II, pp. 13451440.
[15] S. Ling and P. Sol, On the algebraic structure of quasicyclic codes I:Finite fields, IEEE Trans. Inform. Theory, vol. 47, pp. 27512760, Nov.2001.
[16] F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward,Self-dual codes over (4), J. Comb. Theory, Ser. A, pp. 288318,1978.
[17] E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook ofCoding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, TheNetherlands: North-Holland, 1998, vol. II, pp. 177294.
[18] R. P. Russeva, Self-dual [24,12,8] quaternary codes with a nontrivialautomorphism of order 3, Finite Fields and Their Appl., vol. 8, pp.3451, 2002.
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[19] V. Pless, A classification of self-orthogonal codes over (2),Discr. Math, vol. 3, pp. 209246, 1972.
[20] V. Tonchev and V. Y. Yorgov, The existence of certain extremal[54 27 10] self-dual codes, IEEE Trans. Inform. Theory, vol. 42,pp. 16281631, Sept. 1996.
[21] H. P. Tsai, Existence of some extremal self-dual codes, IEEE Trans.Inform. Theory, vol. 38, pp. 18291833, Nov. 1992.
[22] J. Wood, Duality for modules over finite rings and applications tocoding theory, Amer. J. Math., vol. 121, pp. 555575, 1999.
New Quasi-Twisted Degenerate Ternary Linear CodesRumen Daskalov, Member, IEEE, and Plamen Hristov
AbstractTwenty six ternary linear quasi-twisted codes improving thebest known lower bounds on minimum distance are constructed.
Index TermsQuasi-twisted codes, ternary linear codes.
I. INTRODUCTION
Let GF (q) denote the Galois field of q elements. A linear code overGF (q) of length n, dimension k, and minimum Hamming distance dis called an [n; k; d]q
code. We simply use the term [n; k]q
code if wedo not wish to specify d. A constacyclic (twisted) shift of an m-tuple(x
0
; x
1
; . . . ; x
m1
) is the m-tuple (xm1
; x
0
; . . . ; x
m2
), 2
GF (q) n f0g. A twisted shift of anm-tuple by p positions is a twistedshift repeated p times.
A code C is said to be quasi-twisted (QT or p-QT) if there existssome integer p such that every twisted shift of a codeword by p placesis again a codeword [8].
The block length n of a QT code is a multiple of p, so that n = mp.A matrix B of the form
B =
b
0
b
1
b
2
b
m2
b
m1
b
m1
b
0
b
1
b
m3
b
m2
b
m2
b
m1
b
0
b
m4
b
m3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
b
1
b
2
b
3
b
m1
b
0
(1)
where 2 GF (q) n f0g is called a twistulant matrix.In [1], structural properties of QT codes are considered and it is
shown (similar to the proof for quasi-cyclic (QC) codes in [17]) thatthe generator matrices of QT codes can be constructed from m mtwistulant matrices (with a suitable permutation of coordinates). In thiscase, the generator matrix G can be represented as
G = [B
1
; B
2
; . . . ; B
p
] (2)where Bi
is a twistulant matrix.The algebra ofmm twistulant matrices over GF (q) is isomorphic
to the algebra of polynomials in the ring GF (q)[x]=(xm ) if B is
Manuscript received January 2, 2002; revised March 18, 2003. This work wassupported in part by the Bulgarian National Science Fund under Contract withthe Technical University of Gabrovo, Gabrovo, Bulgaria.
The authors are with the Department of Mathematics, Technical Uni-versity of Gabrovo, 5300 Gabrovo, Bulgaria (e-mail: [email protected];[email protected]).
Communicated by S. Litsyn, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2003.815798
mapped to the polynomial, d(x) = b0
+b
1
x+b
2
x
2
+ +b
m1
x
m1
,
formed from the entries in the first row ofB. The di
(x) associated witha QT code are called the defining polynomials [7], [8]. If = 1, weobtain the algebra ofmm circulant matrices [12], and a subclass ofQC codes . If p = 1 then we obtain codes, which we call twisted (T)codes . If = 1 and p = 1 then we obtain a subclass of well-knowncyclic (C) codes .
If the defining polynomials di
(x) contain a common factor which isalso a factor of xm, then the QT code is called degenerate [7], [8].The dimension k of the QT code is equal to the degree of h(x), where[16]
h(x) =
x
m
gcd(x
m
; d
1
(x); d
2
(x); . . . ; d
p
(x))
: (3)
If the polynomial h(x) has degreem, the dimension of the code ism,and (2) is a generator matrix. If deg(h(x)) = k < m, a generatormatrix for the code can be constructed by deletingm k rows of (2).
Let the defining polynomials of the codeC be in the following form:d
1
(x) = f
1
(x)g(x); d
2
(x) = f
2
(x)g(x); . . . ;
d
p
(x) = f
p
(x)g(x) (4)whereg(x)j(x
m
); g(x); f
i
(x) 2 GF (q)[x]=(xm )gcd(f
i
(x); (x
m
)=g(x)) = 1
anddeg f
i
(x) < m deg g(x)
for all 1 i p. Then we obtain a degenerate QT codes, which,by analogy with one-generator QC codes, are called one-generator QTcodes and for these codes n = mp; k = m deg g(x).
Known results regarding the one-generator QT codes are as follows(see [1]).
Let be such that it does not have an nth root in GF (q). Also, letthe polynomial xm not have multiple roots. The roots of xm are ; ; 2; . . . ; m1, where is a primitive nth of unity and
m
= .
Theorem 1: LetC be a one-generator QT code over GF (q) of lengthn = pm. Then, a generator g(x)g(x)g(x) 2 (GF (q)[x]=(xm ))p of C hasthe following form:g(x)g(x)
g(x)
= (f
1
(x)g
1
(x); f
2
(x)g
2
(x); . . . ; f
p
(x)g
p
(x))
where gi
(x)j(x
m
) and (fi
(x); (x
m
)=g
i
(x))=1 for all 1 ip.
Theorem 2: LetC be a one-generator QT code over GF (q) of lengthn = pm with a generator of the formg(x)g(x)
g(x)
= (f
1
(x)g(x); f
2
(x)g(x); . . . ; f
p
(x)g(x))
whereg(x)j(x
m
); g(x); f
i
(x) 2 GF (q)[x]=(xm )and(f
i
(x); (x
m
)=g(x)) = 1
for all 1 i p. Then p:(d + 1) d(C), where fi: s i s+(d1)g are among the seros of g(x) for some integers s; d(d > 0)and the dimension of C is equal tom deg g(x).
The following theorem will be used to construct new codes fromgiven codes.
Theorem 3 [19]: Let C be an [n; k; d]3
code. If d 2 (mod3)and no codeword of C is of weight 1mod3, then C can be extendedto a self-orthogonal [n + 1; k; d + 1]3
code.
0018-9448/03$17.00 2003 IEEE
Index:
CCC: 0-7803-5957-7/00/$10.00 2000 IEEE
ccc: 0-7803-5957-7/00/$10.00 2000 IEEE
cce: 0-7803-5957-7/00/$10.00 2000 IEEE
index:
INDEX:
ind: