cubic binary self-dual codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 9, SEPTEMBER 2003 2253

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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.


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


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


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


(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.


[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:

cubic binary self-dual codes

Documents