486 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34. NO, 3, MAY 1988

On the [24,12,10] Quaternary Code and Binary Codes with an Automorphism

having Two Cycles W. CARY HUFFMAN

Abstract-A general decomposition theorem is given for codes over finite fields which have an automorphism of a given type. Such codes can be decomposed as direct sums of subcodes which may be viewed as shorter length codes over extension fields. If such a code is self-dual, sometimes the subcodes are also. This decomposition is applied to prove that the self-dual [24,12,10] quaternary code has no automorphism of order 3. This decomposition is also applied to count the number of inequivalent [2r , r ] and [2r + 2, r + 11 self-dual binary codes with an automorphism of prime order r .

I. INTRODUCTION

E EXAMINE linear codes C over finite fields which W have a permutation automorphism. In Section I1 we describe how to decompose C as a direct sum of several subcodes using this automorphism. These subcodes can be viewed as shorter length codes over extension fields of the original field. If C is self-dual, under certain conditions the subcodes are also self-dual. These results generalize the work in [7], [SI, and [15].

In Section I11 we use the results of Section I1 to show that there does not exist a permutation automorphism of order 3 of a self-dual [24,12,10] quaternary code. This, together with [4] and [5], shows that the permutation automorphism group of such a code is a 2-group, if the code exists. In Section IV we examine self-dual binary codes of length 2r or 2r +2, where r is prime. Under certain circumstances, we show how to determine all in- equivalent codes of such type which have an automor- phism of order r . General references for the terminology of coding theory used herein are [ l l ] and [14].

11. THE CODE DECOMPOSITION

Let F9 be the finite field of order q and characteristic p . Suppose r is a positive integer relatively prime to p. Let R = F9[ X I / ( X' - l), where X is an indeterminate. Let J = ( l - X ) and Q = ( l + X + X ' + - 0 - + X ' - ' ) be the ideals of R generated by 1 - X and 1 + X + X 2 + . . . + X r - l , respectively. Let mo( X ) = 1 - X and 1 + X

Manuscript received August 6, 1986; revised August 6, 1987. This paper was presented in part at the 830th Meeting of the American Mathematical Society, Denton, TX, October 31-November 1, 1986.

The author is with the Department of Mathematical Sciences, Loyola University, Chicago, IL 60626.

IEEE Log Number 8821820.

+ X 2 + . + X ' - l = ml( X ) . . . mg( X ) , where m,( X) is irreducible over F9 for 1 I j I g. Let i , ( X ) =

(1 - X r ) / m , ( X ) for 0 I j I g. Let Z, = (i,( X ) ) be the ideal of R generated by i , ( X ) .

Lemma 1: It holds that a) R = Z,,@Zl@ - . . elg;, b) I, is a field for 0 I j I g; c) I,Zk= { 0 } , d) Q = Z o and J = Z l @ - - a $Ig; e) Q = { a ( l + X + X ' + 0 . . + X ' - ' ) ~ U E F , } = F ,

if j # k;

where the map +: Q + F 9 given by ( a ( l + X + X 2 + ... + X'- '))+ = ar is an isomorphism;

f) the identity of Q is (l/r)(l + X + X 2 + . . . + X r P 1 ) ; g) the identity of J is ( - l / r ) ( ( l - Y ) + X +

h) multiplication by (- l /r)( l + (1 - r ) X + X 2 + + X'- ') in J corresponds to multiplication by X

X 2 + . . + X'- ') , and

. modulo ( X ' - 1).

Proof: Parts a)-d) follow from the fact that R is a semisimple ring because r and q are relatively prime, implying that the roots of 1 - X' are distinct. Parts e)-h) are proved in [8, lemmas 2 and 31.

Let C be a linear code over Fq of length n and dimen- sion k; that is, C is a k-dimensional subspace of F;. The weight of a vector x in F; is the number of nonzero entries in x. The minimum distance of C, denoted by d , is the minimum nonzero weight of all the vectors in C. C is called an [ n , k ] or [n , k , d ] code. Let a be a permutation of the coordinates of F t . If x E F; has ith coordinate x,, define (xu), = xfO-l. Such a permutation a is an automor- phism of C , provided xu E C for all X E C . For the re- mainder of this section we assume C has such an automor- phism, which also has only c r-cycles and f fixed points for some r relatively prime to p . Denote the r-cycles by 9,; ., QC and the fixed points by 9c+l,. . e , 9,+!. If x E F;, let xlQ, be the restriction of x to 9,. In a natural way, we may view xisz, as an element a, + a l X + * . + U ~ - ~ X ' - ' of R when 1 I i I c. Furthermore, xalQ8 is (ao + a , X + . . . + U , - ~ X ~ - ~ ) X E R.

Let C( a) = ( x E Clxa = x } and E( a) = ( x E Clxl,, E J for 1 s i I c, xlQ, = 0 for c + 1 I i I c + f }. Both C(u) and E( a) are o-invanant subspaces of C , and C = C( a) @ E(a) by [8, theorem 11. For l s j s g , let E,(a) =

Oa18-9448/88/0500-0486$01.00 01988 IEEE

HUFFMAN: [24,12,10] QUATERNARY CODE AND BINARY CODES WITH AUTOMORPHISM 4x7

{ x E E(a) lx l , , E I , for 1 I i I c}. Let 1 = Cf,,e,( X ) be the decomposition of 1 into primitive idempotents with e l ( X ) E I,. By Lemma If) and g), eo( X ) = ( l / r ) ( l + X + X 2 + . . . + X r P 1 ) and X : = , e , ( X ) = ( - l / r ) ( ( l - r ) + X + X 2 + . . . + X r - ' ) . The following generalizes [15, lemma 3a], which assumed F, = F2 and r to be a prime.

Lemma 2: We have C = C(a) @ E(a). In addition, E(a) = E,(a)@ . . . $E,(a) and E,(a) is a-invariant for

Proof: C = C ( a ) @ E ( a ) by [8, theorem 11. Let U E

E(a). Then by Lemma lg), u = ( - l / r ) ( ( l - r ) + X + X 2 + . . . + X r - ' ) u = X : = , e , ( X ) u . By Lemma IC), e , ( X ) u E E,( a). Therefore, E( a ) = E,( a) + . . . + E& a). Let u E

E,(a)nC,+,E,(a). Then for l s i s g , u l g , € I,n,XA+,ZA = ( 0 ) by Lemma la), so E(a) = E,(a)@ . . . @E,(a). As I, is an ideal of R , E,(a) is a-invariant.

Suppose we have the inner product (., .) on F; of the form

l s j s g .

n

( u , u ) = u,u,p"' (1) i = l

where u, u E F; with u = ( u,; . ., u,?) and u = ( u l ; . . , u,?). Define C = { u E F;l(u, u ) = 0 for all u E C}. C is self- orthogonal under (1) if C c C

Extend the map +: Q + F9 of Lemma le) to @: Q' x Fy' + F:'f where

and self-dual if C = C '.

(a , ( l + x + X 2 + . . . + x r - l ) , . . . ,

a,( 1 + x + x 2 + . . * + x r - l ) , a,+ 1,. . . , a '+ / ) @

= ( a , r , . . . , a ' r , a , + l , . , a ,+ / ) .

Then @ is an F9-vector space isomorphism. By viewing C(a) as a subset of Q'X F;, we may view C(u)@ as an F,-subspace of F:'f. In addition, we can view E,(a) as a code over 1, for 1 I j I g. Let E(a)* and E,(a)* be the codes E(a) and E,(a) where the fixed points ai +l , . . ., are deleted and their codewords are viewed as c-tuples from J'.

Theorem 1: Assume C is a self-dual [ n , n / 2 , d ] code under (1). Then C(u)O is a [ c + f , ( c + f ) / 2 , d ' ] code where d' 2 d / r . If either r 1 (modulo p ) or f = 0, C(u)@ is self-dual under (l), with c + f in place of n .

Proof: This result is in [8, theorems 2 and 31 under a restricted form of (1). Their proofs together with [l, theo- rem 3.21 which was used to prove [ 8 , theorem 21, generalize to the inner product in (1). See also the proof of Theorem 3 to follow.

The next two results generalize [8, theorem 41 and [15, theorem 31, which assumed 1 + X + X2 + + X'-' to be irreducible and m = 0 in (1).

Theorem 2: Suppose there is a nonnegative integer t with q'pm = - 1 (modulo r ) . Then E ( u ) is self-orthogonal under (1) if and only if for 1 I j I g, E,(a)* is self-

orthogonal under the inner product (., .) given by <

( u , u ) = urup'p"' ( 2 ) 1 = 1

where u = ( u , , . . . , u , ) and u = ( u , ; . . , L ~ ~ ) with u , , u r E Z l .

a = ( u , , ~ , a , , , ; . . , a , , , - , ; . . , u , , ~ , a,, , , . . . ,a , , . - , ,O,O; . . ,O) and b = ( b , , , , b , , , ; . . , b l , r - l , . . ., bl,O, b , , l , . . ., b c , r - l , 0,O; . . , O h where a, b E

E ( a). Then as aa E E( a),

Proof: Let

, for all 0 I h I r -1

( 3 )

where the second subscript is read modulo r . Let a*, b* E E(a)* where a* = (C;:\a1,,Xk;. . ,CL:\uL,,XA), h* =

(,X;:\b1,,Xk; . . ,C;lkb, . ,Xk). Then

i r - 1 1 [ ril 1 4 ' ~ " '

( a * , b * ) = 1 1 a l , , X k k , X J r = l , = o J = o

as b,,, E F,. As q'p'" = - 1 (modulo r ) and again reading the second subscript modulo r .

\ \

l = l \ j = O \ h = /

Thus r - 1

( a * , b*) = (ash, b ) X - h h = O

by (3 ) .

( 4 )

Clearly (a * , b*) = 0 if and only if (ash, b ) = 0 for 0 5 h - < r - 1 by (4 ) . If a* E E,(a)*, b* E E,(a)* with j # k , (a * , b*) = 0 by Lemma IC). The result is now clear.

Theorem 3: Suppose a nonnegative integer t exists with 4%" = -1 (modulo r ) . Suppose that C is a self-dual [ n , n / 2 ] code under ( 1 ) . Then for 1 I j I g , E,(a)* are self-dual [ c , c / 2 ] codes under (2) . Conversely, if for 1 I j s g, E,(a)* are self-dual [ c , c / 2 ] codes under (2 ) and C(u)@ is self-dual under (1) with c + f in place of n where r 1 (modulo p ) or f = 0, then C is a self-dual [ n , n / 2 ] code under (1) .

488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO, 3, MAY 1988

Proof: Assume C is a self-dual [ n , n/2] code under (1) over F = Fq. By Theorem 2, EJ(u)* is a self-orthogonal [e , d,] code over I, where

C dim[,E,(o)*=d <-, f o r l l j l g ( 5 )

J - 2 as ( a , a ) is nondegenerate over I,. By Lemma 2 and Theo- rem 1,

n - =dimFC=dim,C(u)+dim,E(u) 2

c + f g = -+ dimFI,dim E ( u ) .

I, J / = I

Hence by (5) , n c + f c g c + f c

2 - 2 2 j = 1 2 2 -<-+- dimFI,=-+-(r-l)

c r + f n 2 2

=-- - -

111. APPLICATIONS TO A [24,12, lo] QUATERNARY CODE

In this section we assume C is a self-dual [ n , n/2] code over F4 under the inner product (1) where p" = 2. Such codes have been classified up to length n =16 in [3] and [lo]. By [lo, corollary 151 the minimum distance d satisfies d I 2[ n /6] + 2. If d = 2[ n /6] + 2, C is called extremal. Length n = 24 is the shortest length where the existence of such an extremal code is unknown (see [3, sec. VI11 and [lo, table I]); in this case d =lo. If this code exists, the supports of codewords of weights 10, 12, and 14 form 5-designs (see [lo, theorem 181).

The automorphism group of C , denoted G(C), consists of all n X n monomial matrices M over F4 such that CM = C. In [4], the only possible primes r dividing the order of G(C) where C is a self-dual [24,12,10] code are shown to be 2, 3, 7, and 11. In [5], 7 and 11 are eliminated by computer. The permutation automorphism group con- sists of the permutations of G(C). In this section, we eliminate a permutation of order 3 without computer assis- tance. We remark that similar work has been done on the possible existence of a [72, 36, 161 doubly even binary code -

in [4], [9], [12], and [13]. For the remainder of this section C is a self-dual

[24,12,10] quaternary code with a permutation automor- phism u of order r = 3. By [4] and [5] , the only case we need to examine is where u is a permutation with c = 8

using Lemma la) and e). Thus all inequalities in the foregoing equations must be equalities, showing that for 1 I j I g , E,( a*) are self-dual [ e, c/2] codes.

Let a , b E C(u) where

a = ( a , ( l + x + x2+ . . . + x r - 1 ) ; . .,

a , ( l + X + X 2 + * * * + X ' - l ) , a L + , , . . , a , + ~ )

and b = ( b , ( l + X + X 2 + e . . + X r - ' ) , - . . ,

I b, (1 + X + X 2 + . . . + X' - ) , b, + . , b, +, ) . Then

c c + f ( a , b ) = ra,b,f+ a,b,""'

I = 1 r = c + l

and L c +f

( a @ , b o ) = r'+P"a,b~"+ a,b,P"'.

If r = 1 (modulo p ) , ( a , b ) = ( a @ , b @ ) . If f = 0, ( a @ , bia) = rp"l(a, b) . These facts, together with the re- verse of the previous argument, give the converse.

The next result generalizes [8, corollary 21 and [15, corollary 21 which again assumed 1 + X + X 2 + . . . + X r P 1 to be irreducible and m = 0 in (1).

Theorem 4: Suppose that C is a self-dual [ n , n /2] code under (1). Suppose there is a nonnegative integer t with q'p" = - 1 (modulo r ) . Then c is even.

Proof: This is a direct consequence of Theorem 3.

I = 1 r = c + l

Lemma 2 and Theorem 3 provide the basic tools neces- sary for the applications of Sections I11 and IV.

3-cycles and f = O fixed points; therefore, we assume u = (1,2,3)(4,5,6). . . (22,23,24). By Theorem 1, C(u)@ is a self-dual [8,4, d'] code where d ' r 10/3. By [lo], C(u)@ must be the [8,4,4] code E,. Hence we may assume C(u) has the following generator matrix where 1 is 1 + X + X 2 :

a, a2 a3 a4 a5 a6 a7 a*

1 . L 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

Let F4 = {0,1, w , a'} where 1 + w + w2 = 0. Over F4, 1 + X + X 2 = ( a + X ) ( w 2 + X ) . Let I,, = Q = (1 + X + X 2 ) , Z l = ( ( 1 + X ) ( w 2 + X ) ) , and Z , = ( ( l + X ) ( w + X ) ) . The fields I,,, I,, and Z2 are isomorphic to F4 with the correspondence given in Tables 1-111.

The condition q'pm = - 1 (modulo r ) of Theorem 3 is satisfied by t = 0 in this case. Hence ,?,(a) = El(u)* and E 2 ( u ) = &(a)* are self-dual [8,4] codes over Z, and I, , respectively, under the inner product (2), which is the same as (1) since t = 0. By examining Tables I1 and 111, we see that a weight z codeword in E , ( u ) or E2(u) , viewed as a vector in F424, corresponds to a weight (1/3)z codeword in El(u)* or E2(u)*, viewed as a vector in Z: or 1:. Hence E,(u)* and E2(u)* are self-dual [8,4,4] codes. By [lo], they also must be the code E*, generated by matrices as in the foregoing matrix with permuted columns.

To determine generating matrices for El(u)* and E2(u)*, we use the following notation, also described in [7]. A duo is a pair of coordinates for a code. A cluster for

H U I - F W N : ( 2 4 . 1 2 . 1 0 1 QlJATERNARY CODb AND BINARY CODrS WITH AIJK)MORPHISM 4x0

TABLE I CORRESPONDENCE BETWEEN I,, AND

0 0 1+ x+ I

w + w x + w x 2 w w? -t 02x+ w 2 x 2 w2

TABLE I1 CORRESPONDENCE BETWEEN I , AND F4

1, F4

0 0 1 + w 2 x + w x * 1 w t x + w’x2 w w2 + w x i- x’ 7

w”

TABLE 111 CORRESPONDENCE BETWEEN I , ANI) F4

1’ F4 0 0

1 t w x + w 2 x 2 1 w + w 2 x + x2 w b? + x + w x ’ 7 w-

a code is a set of disjoint duos such that the union of two distinct duos forms the support for a weight-4 vector in the code. A d-set for a cluster is a subset of coordinates such that there is precisely one element of each duo in the d-set. A defining set for a code will consist of a cluster and a d-set provided the code is generated by weight-4 vectors arising from the cluster and the vector whose support is the d-set. Clearly, E, has a defining set; for example, the defining set for C(a)@ has duos { Q l , Q 2 } , {52,,Q2,}, { Q5, a,}, {a,, Q , } , and d-set { Q, , Q , , as, Q , } from the generator matrix for C(a).

We wish to determine defining sets for El(o)* and E2(a)* which will not lead to vectors in C of low weight. By examining Tables 1-111, it is clear that a nonzero element of Z, added to a nonzero element of Z,, with i # j , leads to an element of R of the form a , + a , X + u 2 X 2 with exactly two of a,, a,, a 2 nonzero. This observation yields the following lemma.

Lemma 3: Let x, E C(a), x1 E El(a), x2 E E2(a) with x,, xl, x2 viewed as elements of F424. Let x:, x; be xl, x2 viewed as elements of I:, 1:. Then two of x,@, x:, x; cannot be weight-4 vectors in C( a)@, El( a)*, E2( a)*, re- spectively, with the same supports.

Proof.. Let x; = xo@. If XI*, x/* are weight4 vectors with the same supports and i # j , then x, + xJ has weight 8 in C , by the previous observation.

We now show that it is impossible to find defining sets for E,(a)* and E2(a)* without obtaining two vectors in C(u)@, El(a)*, and E2(u)* of weight 4 with the same supports, violating Lemma 3. The coordinates for the defining sets are { Q1; . ., a,}. For simplicity denote Q , by

TABLE IV

i . We order the coordinates so that the generator matrix for C( a ) is given as before. By [lo, table IV] it is clear that the automorphism group of E, is triply transitive. Hence a cluster for E, can be chosen so that any pair of coordinates for the E, forms a duo. Therefore, we assume { 1,2}, { 3, c} are duos for E,(a)*. If c = 4, vectors in C(u)@ and El( a)* exist with supports {1,2,3,4}, violating Lemma 3. Clearly, (5,6)(7,8) and (5,7)(6,8) are in the automorphism group of C(a)@; therefore, we may assume c = 5 . As (1,2)(7,8) is an automorphism of C( a)@, the remaining two duos of El(a)* are {4,6}, (7,s) or {4,7}, {6,8}. If they are {4,6}, {7,8}, vectors in C(a)@ and El(a)* exist with supports { 1,2,7,8}, violating Lemma 3. Therefore, the cluster of El(a)* is {1,2}, {3,5}, {4,7}, {6,8}. Note that once the cluster is chosen two different d-sets which differ in an even number of duos define the same code. Hence the possible d-sets for El(a)* are {1,3,4,6} or {1,3,4,8}. If the d-set is {1,3,4,6}, then C(u)@ and El( a)* possess vectors with supports { 1,4,5, 8}. Therefore, the d-set for El(a)* is {1,3,4,8}.

We may assume {1,2} is a duo of E2(a)*. As C(a)@ has vectors with supports { 1,2,3,4}, { 1,2,5,6}, and {1,2,7,8}, then {3,4}, { 5 , 6 } , and {7,8} cannot be duos of &(a)*. As El(a)* has vectors with supports {1,2,3,5}, (1,2,4,7}, and {1,2,6,8}, then {3,5}, {4,7}, and {6,8} cannot be duos of E2( a)*. The possible clusters for E2( a)* are, therefore, 1) {1,2}, {3,6}, {4,8}, {5,7}; 2) {1,2}, {3,7}, {4,6}, { 5 , 8 } ; 3) {1,2}, {3,8}, {4,5}, ( 6 7 ) ; or 4) {1,2}, {3,8}, {4,6}, {5,7}. Each of these four cases has two possible associated d-sets. Table IV lists the possibili- ties; the column headed “support” gives the support of a weight-4 vector in E2( a)* and the column headed “C( a)@ or El( a)*” indicates which subcode also contains a weight- 4 vector with the given support. In all cases Lemma 3 is violated. Thus we have the following theorem.

Theorem 5: There does not exist a self-dual [24,12,10] quaternary code with a permutation automorphism of order 3. The permutation automorphism group of such a code is a 2-group.

Proof: This follows from the foregoing together with [4] and [5].

We remark that if r = 3, 11, or 19 is the order of a permutation in the automorphsm group of a self-dual [ n , n/2] code, then c must be even by Theorem 4. (Here t = 0, 2, or 4, respectively.) This gives an alternate proof to part of [4, theorem 71.

490 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO, 3, MAY 1988

Iv. CODES WITH AUTOMORPHISMS HAVING TWO self-dual code under (2) generated by

[ e , ( X ) a , ( x > l (8) In this section, we examine binary self-dual codes of where e,(x) is the identity of 1,

length 2r or 2r + 2 where r is a prime, possessing an automorphism having two r-cycles. Such codes, by correct e, ( X ) + a , ( x )2' + = 0 , (9) ordering of the columns, can be seen to be quasi-cyclic (see b 111, ch. 161). Quadratic residue codes and the Pless SYmme- With the association between I,,, and c$) as before,

the decomposition results of Section I1 and a nice result of By Lemma 4a), as tIcp = q), fE , (ao (x ) ) generates z z l . Yorgov (Lemma 6 to follow), Theorem 7 gives a count of Let a,( X) = &(ao( X ) ) generate I;"+l for 0 I i I g - 1. Let the number of inequivalent codes of these types, under P,( X ) = a,( X ) 2 -'. By (9), a,( X ) is some power of P I ( X ) , assumption of the existence of the integer t described next. By following the proof, one could, in fact, construct these need information about the decomDosition

r-CYCLES

for 1 I i I g

Y (2).

try codes Over F3 are examples of quasi-cYclic 'Odes. Using let 1: be generated by (yo( X ) as a multiplicative group.

and any power of satisfies (9). we later

codes. The notation of Section I1 will be used, where p = q = 2

and p"' =1 in the inner product (1). We also assume that r > 2 is a prime such that there exists a nonnegative integer t with 2'= -1 (modulo r). We assume t is the smallest such integer. Let Z, be the integers modulo r, and let 5 be a generator of Z," = Z, - {0} such that (g = 2 (modulo r). The cyclotomic coset C,(2) modulo r is { 4 2 4 . . .,22'-1i} G Z,. Because r is a prime, Cd2) = {0} and C,(2) has order 2t if i f 0 (modulo r). The distinct cyclotomic cosets are Cd2),Ci:), C$), , C$l, where 2tg = r - 1. Each of these cyclotomic cosets is associated with a field Io, Z1; . *, Zg of Section I1 as follows. Io = F2 is associated with CJ'); Z,+l is associated with Ci?, where the roots of m , + , ( X ) are { v'll E C$)} and v is a root of X ' - 1 in a splitting field over F2. Therefore, Z, = Z2 = . . . = Zg = F22,.

Let f s r : R + R , where f e ( p ( X ) ) = p ( X c ' ) . This is clearly a well-defined ring isomorphism because r is prime. Fur- thermore, we have the following.

Lemma 4: We have the following:

a) f [ , : Io - Io and f e : I,+, + Zk+ 1, where <'C$) = C$) are isomorphisms;

b) f c f p = fe.1; and c) Z," = { f e 10 I i I r - 2) under the map 5' + f c .

Proof: By Lemma 1, it is clear that &: Io + Io is an isomorphism as r is a prime. If p ( X ) # 0 and p ( X ) E Z,+ the nonzeros of p ( X ) are { v'lf E C:,')}. The nonzeros of p ( X * : ) are { ( V ' ) ~ I ~ E C$)} = { v 'J1 E CP)}. Therefore, p ( X s ) E Zk + and a) is clear. Also b) is obvious and c) follows from b).

Let C be a self-dual binary code of length 2r or 2r + 2 and a an automorphism of order r with c = 2 r-cycles. By Lemma 2, C = C( a) $E,( a) @I * $E,(a). If C is a [2r, r] code, a generator matrix for C(a) is

[1 11 ( 6 ) where 1 = 1 + X + X 2 + e - - + Xr-'; if C is a [2r +2, r +1] code, a generator matrix for C(a) is

where the r-cycles are on the left and the fixed points on the right. (Blanks indicate zero entries.) Also Ei(a)* is a

of X . X = Cf=,Xe,( X ) = Cf==,y,( X ) , where y,( X ) = Xe,( X ) E I,. Note that Xe,( X ) # 0, and Xe,( X ) # e,( X ) when i # 0. Hence as X' = 1 = C,P_,y,( X) ' , by Lemma IC), y,( X ) is an element of order r in the multiplicative group I;" for l s i s g . As (2'+1)(2'-1)=22'-1, and 2 ' ~ - 1 (modulo r), yJ+l( X ) = PJ( X) ' ) , where

We may rechoose a o ( X ) if necessary so that co= (2' + l ) / r . If we do that, we have the following.

Lemma 5: We have

c j = ( 7 ) 5 - ' ( r n o d ( 2 ' + 1 ) ) , f o r O I j I g - 1

where 5-J is the inverse of [ J in Z,",

ZfL$,( X ) ' Z [ ~ by Lemma IC). However

g-1

Proof: As X = y o ( X ) S C ~ ~ - , ' p , ( X ) ' ~ , X E J = y 0 ( X ) [ ' +

x 5 J = f [ J ( = f [ J (YO ( x ) ) + c f ( J ( Pi ( )) " r = O

by Lemma 4a). Since

= ( CYJ (x)2'-') ' O = PJ( x ) '0 E I,+ 1 ,

for 0 5 j I g - 1, pJ ( x ) 'J' = p, ( x ) ' O .

Therefore, cJ5J = co (modulo (2t + 1)). Now 5-JtJ = 1 (modulo r), which implies that

( 2 7 - [ - J [ J = ( 2 ' 3 - (mod(2'+1)).

By (lo), cj

We wish to count the number of different choices for ai( X) in (9) which lead to inequivalent codes. The follow- ing lemma, found in [17], will be useful.

Lemma 6 [17, theorem 61: Let u be an automorphism of prime order r of two self-dual binary codes C and C', as described in Section 11, where r > (c + /+4f)/2.

cot-' (modulo (2' + 1)).

HUFFMAN: [24,12,10] QUATERNARY CODE AND BINARY CODES WITH AUTOMORPHISM 491

Suppose every automorphism of order r of C and C’ has Replacing p ( X ) by this Xpp( X ) , we may assume 0 I bo < the same number of r-cycles and fixed points as a. Then C (2‘ + l ) / r . It is clear that r( p ( X ) ) = r( q( X ) ) or and C’ are equivalent if and only if C’ can be obtained T ( p ( X ) ) n r ( q ( X ) ) = 0 for q ( X ) E A. Hence rA =

from C by application of a product of some of the follow- ing transformations to C = C( a) @E(a): 5‘2, (modulo r ) with 0 I i I g - 1,

a) the substitution X + X A for 1 I X I r - 1 in E( a)*;

{ I?( p ( X ) ) l p ( X ) = X p p ( X’) for some p E Z, }.

0 I j I 2t - 1. Then p ( X ) = Xpp( X’) if and only if Lemma 7: Let A

b) multiplication of the j t h coordinate by X ~ I , where

c) permutation of the r-cycles of a; or d) permutation of the fixed points of a.

0 ~ p , I r - 1 , l s j s c;

We remark that in our case c = 2 and f = 2 and that ( 2 + J22+4.2)/2 < 3. Also by Theorem 4, C and C’ cannot have an automorphism with one r-cycle. Therefore, the conditions of Lemma 6 are satisfied in our case. To each {a,( X ) ; . ., a , ( X ) } satisfying (9), we associate p ( X ) = a,( X ) + . . . + ag( X ) E J . By (9), each a,( X ) must be invertible in I,, and p ( X ) is invertible in J . Also permut- ing the two r-cycles changes the generating matrices to

[ a , ( x ) e,(x>l which is equivalent to

[ e , ( x ) (.,(x))-’].

.F ( p ( x ) ) = { x p p ( x A ) 7 ( x p p ( x”) - IP E 4, E z:> .

Thus the polynomials in J which lead to codes equivalent to C are precisely those in the set

Let

1 g - 1 A = p ( x ) ’ Pk (X)hk lbkEz2‘+1 f o r O I k I g - 1 . { k = O

As P k ( X ) has order 2‘+1 and Zk has characteristic 2, ( P k ( XA)) - ’ = Pk( Xx)2‘= P k ( X”‘). Thus if p ( X ) E A, ( X p p ( X x ) ) - ’ = X - ” p ( X ” ) ; hence 9 ( p ( X ) ) =

{ Xpp( X A ) l p E Z,, X E Z:}. For p ( X ) E A, let r( p ( X ) ) = { Xpp( X ) l p E Z, }. As r is prime,

f A ( r( P ( X > 1) = { f x ( X p p ( X I ) I P E Zr 1 = { X ” p ( X X ) I p E Z r }

= { X p p ( X A ) ) I p E Z r } =r ( ~ ( x ’ ) ) . Hence

g ( p P ( x ) ) = { f A ( r ( P ( X ) ) ) I X E z : } . Let G = { f , lX~z ,?} and r = { T ( p ( X ) ) l p ( X ) E A } . By Lemma 4c), G is a group isomorphic to ZT. Define r( p ( X ) ) - r( f A ( p ( X ) ) ) for some f A E G. By Lemma 4, - is an equivalence relation on r. = r ( p ( X ) ) ) . Fix p ( X ) = Z : & I $ k ( X ) h k ~ A . By Lemma 5 , there is a unique element

We compute I r / 1. Let r X = { r ( p ( x ) ) l f A ( r ( p ( x ) ) )

g-1

X p p ( x , = P k ( x , hk E r( p ( x ) ) , k = O

2 ‘ + l r

where 0 I bo + pco < -.

i f O < k < g - i - 1 (11) and

b k 2 / + l + (-(‘+*-.)pi 2 ‘ + l = b r + k - g (mod(2‘+1)),

if g - I I k I g -1. (12) Proof: We have

g - 1

p ( x x ) = f A P k ( x ) b k = f A ( P k ( x ) ) h A . [ill k = O

However, f A ( P k ( = ft’2’ ( f t k ( PO ( )

=fC‘+k2,(Po(X)) =Po( xt“k2’)

= Po( x-+k)2’ = f-+k ( Po( x))” by Lemma 4b) and the fact that J has characteristic 2. If 0 I k I g - i -1 ,

fp+’ ( PO ( x))” = Pi + k ( x)2’

As t g = 2 (modulo r), if g - i s k l g - 1 ,

f,,+, ( P o ( XI)” = f&2( Po( x))2’ = ftf+1-.( Po( x))”” - - Pl+k-g(X)2’+1.

Therefore, g - l - 1 g -1

p ( x x ) = P i + k ( x ) b k 2 J + P , + k - g ( X ) h k 2 ” 1 k - 0 k = g - i

However, g-1

x p = y O ( x ) p + P k ( x ) c k p k = O

g - 1 - 1

k = O = YO ( x , I-L + Pi + k ( x , “+”

g - 1

k = g - i + Pi+k-g(X)Er tk-gp .

By Lemma IC) g - 1 - 1

X @ p ( X A ) = Pi+k(X)bk2’+c,+kp k = O

+ g-1 X)hh2’+’+c1+A-~p.

k = g - i

The result follows from Lemma 5.

492 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 3, MAY 1988

We show how to solve (11) and (12). Inductively define

2b,=b,-, (mod(2'+1)), f o r g s k . (13)

Thus (18) is equivalent to

bg, b g + l , . as follows: bk(2('+g/)/d - 1)

= - t P k x p - (2('+gJ)/d - 1) (mod (2' + 1)) ( 2':1) This is well defined as 2 and 2' + 1 are relatively prime. Lemma 8: Equations (11)-(13) are equivalent to which is equivalent to

bk2i+,$-('+,)p - bi+k (mod(2'+1)), ( 2':1)

if 0 S k (14) where s = (2' + l)/gcd(2' + 1,2('+gJ)/d - 1). Notice that

and (13). Proof: Assume (11)-(13) are true. Then (14) is true if bo= - x p ( q) (mods) (20)

0 I k 2 g - i - 1. If g - i I k 2 g then (13) imp1ies and hence (19) is to (20) and 2bk2J + 5-("k)<gp((2' + l ) / r ) = 2b,+, (modulo (2'+1)).

To count the number of bo, b,,. . ., bg-l which solve (11) p( T) = 2( l) (mod(2'+1)), and (12) of Lemma 7, we first count the number of pairs

(bo, p) with 0 I bo < (2' + l ) / r and p E Z, such that (20) yielding (14) for g - i I k I g - 1. For the remaining Val- holds. Denote the count by N(bo, p ) . ues of k , (14) holds by induction. Reversing the argument Lemma 9: We have N(bo, p) = gcd(2' + 1,2(r+gJ)/d -1). gives the converse. Furthermore, if 2('+gJ)/d = 1 (modulo r ) , the set of N( bo, p)

pairs (bo, p) consists of ( l / r ) N ( b o , p) different bo each paired with all possible p E Z,. If 2('+gJ)ld $1 (modulo r ) , the set of N(bo, p ) pairs all have distinct bo.

h = O Proof: Suppose 2('+g')/d = 1 (modulo r ) . Then

= bk+'/ (mod(2'+1)). (15) r(gcd(2' + 1,2('+gJ)Id - 1). Hence sK2' + l ) / r and (20) be-

As 5 g = 2 (modulo r ) , bk E (-kbo (mod 3). (21)

2 '+1 2'+1

By induction, (13) and (14) imply for 12 1

2 '+1 '-1 b k ( 2 l ) ' + [ - (k+" )p ( 7) ( 5'2')

comes bo = 0 (modulo s) for all p E Z,. Therefore, Let d = gcd(i, g ) and S(1) = ((2' + 1)/r)Xi2,-,(5'2J)h. When 1 = g/d, ( 1 / r ) W o , P) = ((2'+ W ) / s

bk(2J)gId + t - ( k + ' ( g / d ) ) p S ( g / d ) - = b k + r ( g / d ) (mod(2' "))- (I6)

As gcd(2'ld,2' + 1) =1, (16) is equivalent to

2'/"bk(2J) ' I d + 2'/"( ( g ) - ' /d( -kpS( g/d)

2'/dbk+g(,/d) (mod(2'+1)). (17)

As 2'/d(5g)-'/d = 1 (modulo r ) ,

2'+1 2 ' + l (mod(2'+1)).

Therefore, by definition of S ( l ) , (17) is equivalent to

b, (2('+gJ)Id - 1) = - [-'pS( g/d) (mod (2' + 1)). (18)

We now assume h = E ' 2 J f l (modulo r ) . Letting x = (1/( h - 1)) (modulo r ) ,

g / d - 1 ( 5 ' 2 9 = ( X g I d - 1).

h = O

= - (2('+gJ)/d - 1 ) x (modr) .

Therefore,

= ( l / r ) g ~ d ( 2 ' +1,2('+gJ)Id-1).

Now assume 2('+g')/d f 1 (modulo r ) . Suppose

( 2 ' + 1 ) ( 2 1 - 3 y = - x p 2 (mods). (22)

We show that (22) holds if and only if p1 = p2 (modulo r ) . If pl = p 2 (modulo r ) , clearly (22) holds. Assuming (22) holds, then -xpl = - x p 2 (modulo u ) where u =

s/gcd(s,((2' + l ) / r ) ) . However, as 2('+gJ)ld f 1 (modulo r ) , gcd(2' + 1,2('+gJ)/d - 1)1(2'+ l) /r. Therefore rls and (s/r)1(2'+ l ) / r . Thus gcd(s,((2'+ l ) / r ) ) = s or s / r . If gcd(s,(2' + l ) / r ) = s, then sI(2' + l) /r which implies rlgcd (2' + 1, 2('+gJ)ld - l), contradicting 2('+gJ)/d f 1 (modulo r ) . Therefore, u = r and - x p l = - x p 2 (modulo r ) . As x f 0 (modulo r ) , p1 = p 2 (modulo r ) . Thus there are r gcd(2' + 1,2('+gJ)ld - 1) choices for bo with 0 I bo < 2' + 1 solving (20). Since bo + ((2' + l ) / r ) is also a solu- tion of (20) if bo is, it is clear that N(bo, p) = gcd(2' + 1, 2('+gJ)ld - l) , and the rest follows.

Let d(i) = gcd(i, g). Let d = { ( I , j ) # (0,O)p I i < g, 0 I j < 2t) . Then

Theorem 6:

S ( g / d ) = ( 7 ) ( 2 ( i + g J ) / d - l ) x 2 ' + l (mod(2'+1)). 1 +- r ( r -1) (2' + 1) g.

HUFFMAN: [24,12,10] QUATERNARY CODE AND BINARY CODES WITH AUTOMORPHISM 493

Proof Suppose h =[ ‘2J f l (modulo r). The sub- scripts {O,l; . ., g -1} = Zg of b, are broken into orbits of size g / d ( i ) by the action k - k + i . For 0 I w I d( i) - 1, the w th orbit { b,, b,+,; . ., bw+(g!d( , ) - l )r }, where the subscripts are read modulo g, is unique once p and b, are determined because (11) and (12) determine b,+,, bw+2,,. . -, b w + ( g / d ( r ) - l ) r . We determine b, from bo and (21). We remark that if k w + ( g / d ( i) - l) i (modulo g), (12) determines the original b, because (11) and (12) can be recovered from (13) and (15); (15) was used to determine b,. If we know bo, (21) yields gcd(2‘ + 1, 2(‘+gJ)/d(r) - 1) choices for each b, with 1 I w I d ( i ) - 1. Thus for each (bo, p ) there are (gcd(2‘+ 1, 2(’+gJ)/d(r) -l))d(‘)-l choices for the b, not in the 0th orbit. Each pair (bo, p ) determines a different 0th orbit as follows. If 2(’+gJ)ld $1 (modulo r), this is clear as differ- ent pairs have different bo, by Lemma 9. Assume 2 ( r + g ’ ) / d

TABLE V

3 1 1 5 2 1

11 5 1 6 1 13

17 4 2 19 9 1 29 14 1 37 18 1 41 10 2 43 7 3

1 1 2 2 2 4

22 202 644

1200

REFERENCES

[l] R. P. Anstee, M. Hall, Jr., and J. G. Thompson, “Planes of order 10 do not have a collineation of order 5,” J . Comhin. Theory A , vol.

N. L. Biggs and A. T. White, Permutation Groups and Comhinu- toriul Structures, London Math. SOC. Lecture Note Series 33.

29, pp. 39-58, 1980. [2]

-1 (modulo r). Note that if i = O , then d(0) = g and 2 J ~ 1 (modulo r ) which only occurs for j = 0 as < 2t. Hence i # 0. Thus the 0th orbit has at least two entries bo and b,. However, if two pairs (bo, pl) and (bo, p2) yield the same b,, using (11) with k = 0, then [-b1((2‘ + l ) / r ) = [-‘p2((2‘+1)/r) (modulo (2‘+l)) which implies [-;U2 (modulo r ) or p1 = p 2 (modulo r ) . Thus

Cambridge, UK: Cambridge Univ. Press, 1979. [3] J. H. Conway, V. Pless, and N. J. A. Sloane, “Self-dual codes over

GF(3) and GF(4) of length not exceeding 16,” IEEE Trans. In- form. Theory, vol. IT-25, pp. 312-322, 1979.

PI J. H. Conway and V. Pless, “On primes dividing the group order of a doubly-even (72,36,16) code and the group order of a quaternary (24.12.10) code,” Discrete Math, vol. 38, pp. 143-156, 1982. -, “Monomials of orders 7 and 11 cannot be in the group of a (24,12,10) self-dual quaternary code,” IEEE Trans. Inform. The-

[5]

OV, VOI. IT-29, pp. 137-140, 1983.

- - (gcd(2‘ + 1 , 2 ( ; + g J ) / d ( f )

If h = 1 (modulo r), then i = j = 0 and any b, solves (11) and p must be 0; so ITl[ = l/r(2‘ + 1)g. As Ir/ - I =

1/( r - l)C, E GII‘xl by Burnside’s lemma ([2, theorem 1.2.51 or [6, theorem 11.3al) the result follows.

Theorem 7: Let r be a prime such that there exists a nonnegative integer t with 2‘ = - 1 (modulo r ) . The num- ber of inequivalent self-dual [2r , r ] binary codes with an automorphism of order r is !I?/ - I. The number of in- equivalent self-dual [ 2 r + 2, r + 11 binary codes with an automorphism of order r is 21r/ - 1. (r/ - I is given in Theorem 6.

Proof: This follows from (6) and (7) together with Lemma 6.

Table V gives the size of (r/ - I for primes r < 50 whenever there exists an integer t such that 2‘ = - 1 (mod- ulo r).

[6] L: Dornhoff, Group Representation Theorv. New York: Marcel Dekker, 1971.

[7] W. C. Huffman, “Automorphisms of codes with applications to extremal doubly even codes of length 48,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 511-521, 1982. -, “Decomposing and shortening codes using autornorphisms,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 833-836, 1986. W. C. Huffman and V. Y. Yorgov, “A [72,36,16] doubly even code does not have an automorphism of order 11,” IEEE Trans. Inform. Theoty, vol. IT-33, pp. 749-752, Sept. 1987. F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, “Self-dual codes over GF(4),” J . Comhin. Theory A , vol. 25.

[ l l ] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes.

[12] V. Pless, “23 does not divide the order of the group of a (72,36,16) doubly even code,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 113-117,1982. V. Pless and J. G. Thompson, “17 does not divide the order of the group of a (72,36,16) doubly even code,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 537-541, 1982.

[14] V. Pless, Introduction to the Theory of Error-Correcting Codes. New York: Wiley, 1982.

[15] V. Y. Yorgov, “Binary self-dual codes with automorphisms of odd order,” Problems Inform. Transmission, vol. XIX, pp. 11-24, 1983.

[16] -, “Doubly-even extrernal codes of length 64,” Prohlems In- form. Transmission, vol. XXIII, pp. 277-284, 1987.

[17] -, “A method for constructing inequivalent self-dual codes with applications to length 56,” IEEE Trans. Inform. Theon.. vol. IT-33,

[8]

[9]

[lo]

pp. 288-318,1978,

New York: North-Holland, 1977.

[13]

pp. 77-82, 1987.