on extremal self-dual quaternary codes of lengths 18 to 28. i

I E F F T K A N \ A C T I O N \ Oh INFORMATION T I I F O R Y , VOI . 36. h0. 3 , M A Y 1990

It is easy to see that block uB + v, i i # 0, intersects a given block XB + y , x f 0 in at least two points if and only if u =

- x . (b , - b , ) / (b , - b,)-acd v = y + x (b , I ( b , b , ) (b l ) / (b l - b,)) for some b, , b,, b , , b, E B, b , # b,, b , # b,. One checks that xb, + y = U $ , + v,xb, + y = u h , + v and xb, + y # xb, + y since b , # b,. Several of these blocks coincide with XB + y .

Theorem 4.1: uB + v = xB + y b , = bb, and b, = bb, for some b E B.

Proof: Suppose that uB + v = XB + y . By Theorem 2.1, v = y andJb, - b,) / (b , - 8,) = b for some b E B. Thus, 6 , + bb, = b, + bb,. From v = y , we get b, = ( b , - b,)b, /(6, - 8,); hence, b,b, = b,b,. Consequently, bb,b, - b , - bb, = bb,!, - b, - bb,. If b, , bb,, b,, bb, are all unequal to 1, then (b , , bb,} = (b,, bb,} by Theorem 3.5. Since b , # b2, this implies b , = bb, and b, =

bb,. If one of these elements is equal to 1, it is easy to check that we get the same result. The converse is straightforward. 0

Next, we determine when two blocks, which intersect XB + y in exactly two points, coincide.

Theorem 4.2: Let b,,b,,h,,b,,c,,c,,C,,C, be elements of B such that b , # b,,b, # b,,c, # c2,C, # C2. Let uB + v, sB + t be the blocks associated with b,, b, and c,, C,, i E (1,2), and suppose both blocks are unequal to xB + y . Then, uB + v =

sB + t b, = cl,!, = c,, E l = bT,, b2 = bC, for some b E B or b , = c,, b, = c , , b , = bC,, b, = bc, for some b E B.

Proof: “ 3’’ We have that xc, + y = sC, + t , xc, + y = sC, + t . Since uB+ v = sB + t , it follows that v = i and s = bu for some b E B. Since sB + t is unequal to XB + y , it intersects XB + y in exactly two points by circularity. Thus, (b , , b,} = { c , , c , } and {b,,b,)={bC,,bc,). If b,=_bT,, b,=bC,, then b , = c , , and b, = c , . In addition, b , = bc,, b , = bC, imply b , = c,, b, = c1.

0

Using 4.1, 4.2, one can use a computer to construct a subcode

The converse follows by straightforward calculations.

g4. I .

V. DECODING

Easy decoding methods are substantial features of “efficient” codes. For the row codes corresponding to gq,,, the first author has developed an algorithm using systems of linear equations, which are similar to the decoding algorithms for BCH codes.

On the other hand, typical codes originating from designs often have numbers of codewords that are not too big. Then, it is easier to match an incoming message m with all codewords c E C . As soon as rn is found to differ from c at more then 1/2(dmin(C)- 1) places, c is “rejected,” and a computer takes the next c’ E C. Table I contains information about d,,, for the designs considered. The (sometimes) unknown parameter p in this table (for column codes) can be estimated by using, say, the well-known Hamming bound. More about decoding \ r i l l appear

forthcoming paper.

REFERENCES E. F. Assmus and H. F. Mattson, “On tactical configurations and error-correcting codes,” J. Coinb. Theory, vol. 2, pp. 243-257, 1967. E. F. Assmus and H. F. Mattson, “Coding and combinatorics,” SIAM Rev., vol. 16, pp. 349-388, 1974. T. Beth, D. Jungnickel, and H. Lenr, Design Tliroty. Mannheim, FRG: Bibl. Inst., 1985. 1. F. Blake and R. C. Mullin, The Marlieinuticul Tlzeow of Coding. New York: Academic, 1975.

65 1

W. G . Bridgcs, M. Hall, and J . L. Hayden. “Codes and designs.” J. Coriib. Tlieory, vol. 31, pp. 15.5-174, 19x1. J. R. Clay, “Generating balanced incomplete block designs from planar near-rings.“ J. Algebru, vol. 22, pp. 319-331, 1972. J. R. Clay, “Applications o f planar near-rings to geometry and combinatorics,” Res. Math., vol. 12, pp. 71-85, 1987. J. R. Clay, “Circular block designs from planar near-rings,’’ Ann. IXscretr Math., vol. 37, pp. 95-106, 1988. J. R. Clay, “Geometric and combinatorial ideas related to circular planar near-rings,” to he published. W. G. Cochran and G. M. Cox, Experiinend Designs, 2nd ed. N e w York: Wiley, 1957. G. Ferrero. “Stems planari e BIB-disegni,” Riv. MuI . Unw. funnu. , vol.

G. Ferrero. “Esperimenti statistici e codici di trasmissione,” Sein. Alg. Geoin., vol. 2, Univ. di Brescia, 1985. R. Lid1 and G. Pilr, Applied Abstruct AIgebru, UTM Series. New York: Springer, 1984. F. J. MacWilliams and N. J. A. Sloane, The Theoty of Oror-Cumectlng Codes, I , / I . M. C. Modisett, “ A characterization of the circularity of balanced incomplete block designs,” Uriliras Math. vol. 35 , pp. 83-94, 1989. G. Pilr, Near-rings, 2nd ed. C. J . Salwach, “Planes, biplanes, and their codes,” Ainrr. Murh. Monlhly, vol. X8, pp. 106- 125, I98 1.

1 I , pp. 79-96, 1970.

Amsterdam: North-Holland, 1977.

Amsterdam: North-Holland, 1983.

On Extrema1 Self-Dual Quaternary Codes of Lengths 18 to 28, I

W. CARY HUFFMAN

Abstract -A general decomposition theorem is given for self-dual codes over finite fields that have a permutation automorphism of a given form. Such a code can be decomposed as direct sum of subcodes that may be viewed as shorter length codes over extension fields where the dual of each direct summand is also a direct summand. We also present situations in which it is easy to distinguish such codes. We then apply these results to enumerate some of the extremal quaternary self-dual codes of lengths 18, 20, 22, 26, and 28.

I. INTRODUCTION

In [7], a decomposition of linear co’des B over a finite field F(, is given when 8‘ has a permutation automorphism of prime order r relatively prime to q. In particular, 8 is a direct sum of subcodes that can be viewed as codes over extension fields of F,. Under certain other restrictions if t is self-dual, these subcodes are also self-dual. In this correspondence, we remove these latter restrictions and show that the dual of any of the direct summands is also one of the direct summands. This more general decomposition theory is presented in Section 11. In Section 111, we examine situations in which the equivalence of two such codes can easily be decided; the maps used to decide equivalence in these situations preserve the decomposition.

In [4] and [12], all self-dual codes over F4 of a length of at most 16 are enumerated. An exhaustive search for higher lengths seems infeasible. Still, one would like to enumerate the extremal codes if possiblc. In [11], it is shown that there does not exist a 124, 12, 101 self-dual code over F4. (See also 121, [3], 171, [9].) In Section IV, we examine the lengths 18, 20, 22, 26, and 28. In particular, we enumerate the extremal self-dual codes that have a monomial automorphism of prime order r 2 5. We also show that no extremal code of length 28 exists with a permutation automorphism of order 3.

General references to coding theory are [13] and [141.

Manuscript received February 8, 1989; revised October 3, 1989. The author is with the Department of Mathematical Sciences, Loyola

IEEE Log Number 8933621. University of Chicago, Chicago, IL 60626

0018-9448/90/0500-0651$01 .OO 01990 IEEE

652 IFFE TKANSAC TIONS O N INFORMATION THEORY, VOL. 36, NO. 3. MAY 1990

11. GENERAL CODE DtCOMPOSlTlON

Let c, be the finite field of order q and characteristic p . Suppose r is relatively prime to p . Let R = F , [ X ] / ( X " - I ) , where X is an indeterminate. Suppose X' - 1 = FI;=,,m,(X), where m,(X) is irreducible over F4 and m , , ( X ) = X - 1 . Let I , = ( ( X ' - l ) / m , ( X ) ) be the ideal of R generated by ( X ' - l ) /m, (X) for 0 I j I g. By Lemma 1 of [7], R = I(l@Il @ . . . @ I n , I, is a field for 0 4 j I g, and I, Ik = (0) if j # k .

Consider the map T,,",~,: R - R given by

r - l

T,,#,~,( r c ' a i X i ) = a:"XXI1' where g c d ( r , u ) = 1. i = 0 I = 0

Lemma I :

1) T , , ~ , ~ is a ring automorphism of R; 2) T,,",~ is a field automorphism of I(l; 3 ) T , , ~ , ~ , permutes I I ; . ., IR and if T , , ~ , ~ , ( I ~ ) = I j , it is a field

isomorphism from I , to Z,.

Prooj5 We simply observe that T,,,.,,, preserves addition and multiplication in R, and because gcd(r, U ) = 1, ker T, ," ,~ = (0). Therefore, 1) holds. As T , , ~ , ~ , ( I + X + . . . + X ' - l ) = 1 + X + . . . + X r - I , 2) follows directly and as 1 + X + . . . + X r - ' =

r n , ( X ) . . . m , ( X ) , T , , ~ , ~ , ( ~ , ( X ) ) = m , ( X ) for some j yielding 3) . U

Let 8 be a linear code over F4 of length n and dimension k . The weight of a vector x E F: is the number of nonzero entries of x . The minimum distance d of 8 is the minimum nonzero weight of all the codewords in 8. t is called an [ n , k ] or [ n , k , d ] code. Let U be a permutation of the coordinates of F:. If x E F," has ith coordinate x i , define ( x u ) ; =x,,-I; U is a permutation automorphism of B provided xu E 8 for all x E t. Assume for the remainder of this section that d has a permutation automorphism (T, which has only c r-cycles and f fixed points ( r still relatively prime to p ) . Denote the r-cycles by RI; . . ,Rc and the fixed points by R c + l , . . . , f lc+f . Let xII1, be the restriction of x to RI for x E F,". For 1 I i I c, xIrL, can be viewed as an element a,, + a , X + . . . + a,- ] X r - ' E R, where xuIl2, is ( a , , + a , X + . . . +u, - ,X ' - ' )XER. Let C ( U ) = { ~ E 6 1 x 0 = x ) , and for 1 I j < c, E j ( u ) = { x E C(xl,, E Z, for 1 5 i

and E j ( u ) are u-invariant and t = C ( o ) @ E , ( o ) @ . . . @E,(o). By definition, we may view E l ( u ) as a code over I,. Let E,((T)* be El(u) , where the fixed points R, + . ., RCtf are deleted, and their codewords are c-tuples from IT. Let E(u)= E l ( u ) @ . . . @E,(u) and

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

- < c and xJn, = 0 for c + 1 I i I c + f}. By Lemma 2 of [7], C(u)

= E , ( u ) * @ . . . @E,(u)*.

n

( u , v ) = U i v p " l (1) r = l

where u,v E F<Y with U = ( u I , . . . , u , ~ ) and v = (v,; . . , v ,~ ) . De- fine t ' L = {U E c;'/(u, v > = o for all v E &. t is left self-orthogonal under (1) if B G G I and left self-dual if B = t '>. Analogously, define 8 = {v E F J ( u , v ) = 0 for all U E G}. t is right self-orthogonal under ( 1 ) if t _c g i K and right self-dual if t= g". If B= t I R = BL'~, B is self-dual under (1) . In what follows, we will deal with left self-orthogonality and left self-duality; analogous results follow in the case of right self-orthogonality and right self-duality. In Exam- ples 2 and 3 at the end of this section and in Section IV, the left and right terms can be dropped as 1c= '- = t ".

For the remainder of this section, we let \ , t be nonnegative integers, where s I m with m as in (1). Choose an integer U

such that p'q'u = - 1 (mod r ) . Define the form (., .> on R' by C

( x , y ) = c X1YP'4'. (2) r = l

Lemma 2: If a,b E E(u) are associated with a * , b* E E(u)*,

Proof: Let a = (al,(,, a , , , , . . . , a l . r - l , . . . , a,., , , a,,]:.., then ( a * , ~, , , , , -~,~,(b*)) = ~ ~ , ~ ~ , ( a o " , b ) X - " .

U , I - ,, 0, 0, . . . ,O) and define b analogously. Then

( a u " , b ) = U ~ , , - , ~ ~ [ Y , f o r a I I O s h I r - 1 ( 3 ) 1 = 1 ' ( ' - I ]=I) 1

where the second subscript is read mod r. Then

( a * , ~ , , , ~ ~ - > , ~ , ( b * ) ) = r = l ( : g b a r , , X k ) ( : ~ 1 l ~ [ y - ' X ~ ' ~ ) ' ' ~ '

The result follows by (3 ) . 0

By Lemma 1, we may define the permutation A on 1; . ., g, where T,,,~,-.,,,(I,)= I,,(,). Let 8,; . ., @/ be the orbits of A .

Lemma 3: E(o) L E(u)l under (1) if and only if E,,(,)(u)* C(T, , ,~~- , u ( E j ( u ) * ) ) L L under (2) for 1 5 j I g .

Prooj5 By Lemma 2, ( U * , T , , , ~ - $ ,,(b*))=O if and only if ( a c r " , b ) = ~ f o r ~ r h ~ r - l . If E ( G ) ~ E ( ~ ) ~ L , ( a o " , b ) = ~ , and therefore, E,(j)(u)* c ( T ~ , ~ - , , ~ , ( E / - ( U ) * ) ) ". Suppose E,(j)(u)* ~ ( T , , , ~ ~ - ~ , ~ ( E ~ ( U ) * ) ) ' ~ . Let a* E E,(,)(u)* and b* E E,(u)*. By supposition, ( ~ * , ~ , , , ~ - ~ , , , ( b * ) ) = 0 if k = j . If k # j , then ZA(j)~pp"r-~,u(Ik)= {O} as T,,-, JIk)= I,,,, # I,,(,). The result is now clear. 0

The next result is the main one of the section. It generalizes Theorem 3 of [7].

Theorem 1: Let s, t be nonnegative integers with s 5 rn. Choose an integer U such that p"q'u = - 1 (mod r). If 8 is a left self-dual [ n , n / 2 ] code under (11, then C ( u ) is a left self- orthogonal [ n , ( c + f ) / 2 ] code under (I), and for 1 ~ i ~ g , E,(,)(a)* = (~,,,,~-~,~~(E;(u)*))~ under (2). Conversely, if C(u) is a left self-orthogonal [ n , ( c + f ) / 2 ] code under (11, and if E,(,,(u)* = ( T , , ~ ~ ~ - ~ , ~ ( E ~ ( U ) * ) ) ~ '- under (2 ) for 1 I i I g, then t is left self-dual under (1). In addition, c is even if lql is odd for some j .

Proof: Suppose t is left self-dual. By Theorem 1 of 171, C ( u ) has dimension ( c + f ) / 2 over Fq. By Lemma 3, E,,,,(o)* c ( T ~ , , ~ - , , ~ ~ ( E ; ( ~ ) * ) ) ' '. Therefore' dim,, E r ( u ) * + dimlA(,) E,,(;)(V)* I c as \ I j \ = /I,,(,)l and ~T, ,~~~-~ , , , (E , (u)J)* =

IFEE T R A N S A C T I O N S O N I N F O R M A T I O N T H E O R Y , VOL.. 36, N O . 3. M A Y 1990 653

IE,(cr)*l. Thus, for 1 _c i I g,

dim/c I -dim/h, , , (4 )

We only need to show equality holds in (4) as (., .) is nondegen- erate on I,. Let I , E for 1 I j I 1. Now

n - =dimFc,8=dim,uC(o-)+dimF4 E ( c ) 2

c + f ' -- - + E E dimFq I,dim,, E l ( ~ ) *

J = ] l € P ,

c + f ' -~ - + E dimFqIIJ dim,, E , ( o ) *

J - 1 I E P,

as II,l= /I,I if a , b E @,. But

E d i m / ( E ~ ( a ) * I E (c-dim/h, , ) E , 4 ( ~ ) ( ' ) * ) I E // I E P,

by (4). As i ranges over @,, so does A ( i ) . Hence, C, E dim,, E,(c)* I I q l c - .El dim,, E,(c)* implying for l s j s l

* I q l c I E P, 2 ' 1 dim, ,E , (o) I-

Clearly, the inequality is strict if 2 .t. Iq\cc; however, by the above, we have

n c + f / - + dimFqIIJ E dim,, E , ( a ) *

2 2 J = l I E I",

as

dimF4 I , = dimF4 I,,l I I € I";

since

dimFc, I , = dimF<, 11, if i E q. Therefore,

n c i f c c r + f n - < - + - ( r -1) = - = - 2 - 2 2 2 2 '

Thus, in all of these equations, there must be equality; in particular, there must be equality in (4) and

E,(,)(C)* = (T / , ,p t -s , l , ( E , ( C ) * ) ) 'Id, for 1 I i 5 g

Conversely, suppose C ( C ) is a left self-orthogonal [ n , ( c + f 1/21 code and for

1 5 i I g , E,(,,(a)* = ( T , , , ~ ~ - ~ , ~ , ( .!?,(a)*)) '. The previous equations, as equality holds in (4), show dim !,, P =

(er + f )/2 = n /2. Therefore, we only need to show P is left self-orthogonal. As C ( a ) is left self-orthogonal, and E ( c ) is left self-orthogonal by Lemma 3, we only need to show if U E C(o) and b E E(u), ( a , b ) = 0. As a E C(u) , a = ( a , , , ) ; . ., a l , r - l , . . . , a c , ( , ; . ~ , a c , r - l , a c + l , ~ ~ where ai , , )= ai,, for 1 I j I r - 1 and 1 I i I c. Let a * = ( ~ ~ , ~ , ( l + X + . . . + X r - ' ) ; . . , a , , , , ( l + X + . . . + X r - ' ) ) ~ I i , and let a' be a with the fixed points deleted. Note that as b is zero on the fixed points, ( urnh, b ) = ( a r u h , b ' ) , where b' is b with the fixed points deleted. By the same calculation as in the proof of Lemma 2:

r - I r - l

( a * , b * ) = ( a k P , h ~ ) X - / l = (aa'1,b)X-l '. / I = 0 I1 = 0

However, as IoI , = (0) for 1 I j I g, ( a * , b * ) = 0. Hence, (uc", b ) = 0 for 0 I h I r - 1. In particular ( U , b ) = 0. There- fore, P is left self-dual.

0

In applications, we attempt to choose s , t so that the map T,,,,,-',~, is as simple as possible. We give a few examples.

Example 1: In Theorem 3 of [7], we assume there is a nonnegative integer d with q'p"' = - 1 (mod r ) . Then, we choose s = rn and U = 1. The map T, ,~~~- , , , , = T ~ , ' is the identity; hence, E i ( m ) * is left self-dual for 1 I i I g, which is Theorem 3 of [7].

Example 2: Suppose P is a binary self-dual code (i.e., p = q = 2, rn = 0). Therefore, s = 0. If there exists a nonnegative integer t with 2' = - 1 (mod r), then U = 1, and hence, T , , , ~ ~ - ~ , ~ , =

T ~ , ~ is the identity. (See [6], [lo], [12], and [16], where applications of this are given.) When 2' f - 1 (mod r), choose t = 0; thus U = - 1 and the map T , , , ~ ~ - ~ , ~ , = T ] . is the map X + X - I .

Example 3: In Section IV, we discuss codes over F4 (i.e., p = 2, q = 4) self-dual under (1) where rn = 1. Let r be an odd prime. Let x be the order of 2 in Z,, the integers modr . If x is odd, 2.4'"-')/2 = 1 (mod r), and we choose s = 1 and t = (x - 1)/2; hence, U = - 1, which leads to the map = T ~ , - I

(i.e., X + X - ' ) . If 4/x, 2O.4"l4 = - 1 (mod r), and we choose s = 0 and t = x14; hence, U = 1, which leads to the map T t n - , =

- 1 (mod r), and we choose s = 1 and t = (x -2)/4; hence, U = 1, which leads to T,,,,~-',~ = T , , ] (i.e., the identity).

As equality holds in (51, if is odd, c is even.

T ~ , ~ (i.e., conjugation). If 21x and 4 .t. x, 2 . 4 ( x p _ 2 ) ~ 4 u = -

111. ON T H E EQUIVALENCE OF CODES

In this section, we examine situations in which two codes can easily be shown to be equivalent or inequivalent.

Define 1 J q ) as the group of all n X n monomial matrices over Fy. A matrix M E &Jq) can be written as M = PD, where P is a permutation matrix (permutation part), and D is diagonal (diagonal part). Define . l , : ( q ) as the semidirect product of A,,(q) extended by Gal(F,), which is the Galois group of Fy over F,,. Linear codes P and 8' of length n are said to be equivalent whenever CY' = B T for some T E c4,*(q) . Note that if T ~ . l , : ( q ) , we write T = M T , where M ~ . k , , ( q ) and T E

Gal(F,) and first apply M and then T . Define G ( E ) = ( M E

A,,(q)j8M= P} and G * ( B ) = ( T E A , ~ ; , * ( ~ ) I ~ T = ~ } ; G * ( P ) is the automorphism group of 8.

For the remainder of this section, we assume P is an [ n , k ] code over F, with a permutation automorphism c of prime order r, where r # p , as in Section 11. We therefore, assume ~ = ( 1 , 2 , ~ ~ ~ , r ~ ~ ~ ~ ~ ( c - l ) r + 1 , ( c - l ) r + 2 , ~ ~ ~ , c r ) . Let Z,. be the integers modulo r . We define several elements and subgroups of l , , * ( q ) . Denote the identity of ,k,T(q) by I . If i E Z, define i , = i (mod r), where 0 s i,. < r. For 1 I h I r - 1, define

654 1ErF TRANS,%( TION\ O N I N F O R M A T I O N THFORY, VOI . 36. N O . 3 , MAY 1990

f, as the permutation in A2(4), where ( sr + i)fA = sr +(hi), for 0 I s 5 c - 1 , 0 I i 4 r -1, and xf, = x for cr + 1 I x I cr + f. Let F = (fA 11 I A I r - 1) = Z," = Z , - {O). The effect of applying f, to C is to replace X by X A in each r-cycle of U. For 01 s I c - 1 , define U, = ( s r + 1,sr + 2 ; . .,(s + 1)r) and Y =

. ~ ; : ~ l (0 5 p, 5 r - 1 for 0 I S I c - 1). Application of an element of 7Y' t simply cycles the entries of the r-cycles separately. Let Zo be the symmetric group on 1, . . . , a . If 4 E Zc, define 4* as the permutation ((s - l ) r + i)4* = ((s4 - 1)r + i) for l i s i c , O i i i r - 1 and x ~ $ * = x for c r + l I x < c r + f . Let Z: = {4* 14 E Xc}. If 4 E C,, define 4' as the permutation x 4 ' = x for 1 I x I cr and (cr + i)4' = cr + k#~ for 1 I i 5 f. Let xj =($'I4 E Sf). Application of an element of X: permutes the r-cycles; applying an element of 2;. permutes the fixed points. Let B ={diag(a,;. . ,a , , ) l asr+l = asr+2 = . . . = a ( 5 + ~ ) r for 0 I S 4 c - l} . Let gr(C) be the diagonal r-elements of G ( % ) . N o t e t h a t ~ ~ : , ( t ) a G ( t ) , ~ r ( 6 ) a G * ( t ) , a n d i f r + 4 -1, gr(8) = { I ) . Final ly , le tM= ( N E A,,(q)l NUN-' E ( c r ) g r ( 8 ) ) .

Lemma 4: Let t have automorphism U of prime order r + p , with r > c . Assume r 'i [F, : F,,] = (Gal(F,)I. Then ( ~ ) 8 ~ ( 8 ) / 8 ~ ( 8 ) is a Sylow r-subgroup of G * ( t ) / g r ( b ) under either of the following assumptions.

1 )

2 )

r > f , r + q - 1 , and all permutations of G ( 8 ) of order r have c r-cycles and f fixed points. r z > n and any element of G * ( t ) , whose permutation part has order r , has permutation part with c r-cycles and f fixed points.

Proof As r + lGal (F,)l, a Sylow r-subgroup of G * ( t ) / 9 r ( 6 ) is contained in G ( t ) / g r ( g ) . The result follows

U

The next result describes circumstances under which two codes with automorphism U are equivalent. It generalizes results in [8] and [16].

from Lemma 1 of [8].

Theorem 2: Let C and 6' have the same automorphism U .

Assume ( ~ ) 9 ~ ( 8 ) / 9 ~ ( 8 ) is a Sylow r-subgroup of G*(B)/L2?r(8). Then, 8 and 8' are equivalent if and only if C' = ~ M T for some M E A' and T E Gal(F,). If r 'i q - 1 or if '14 - 1 and q(8 ) c 9, then ,.I.'= WC;Z:LZ *F, where 8 * = { D 1 D is diagonal, and U-IDCTD-' E LZr(8)).

Proof: Assume C' = %T for some T E L , f ( q ) . Then, G * ( 8 ' ) = T - l G * ( t ) T . As U E G*(C'), TuT- ' E G * ( t ) . Hence, (u)LZ,.:I(t) and ( T u T - ' ) ~ ~ ( ~ ) are Sylow r-subgroups of G * ( 8 ) . Therefore, there is an SEC*(&) such that S ( T U T - ' ) S - ' ~ ~ ( G ' ) = ( ~ ) g ~ ( & ) . Let M * = ST. Therefore, M * u M * - ' E (u)gr(t) . Let M * = M T , where M E L J q ) , T

Therefore, M E I 9. In addition, 8' = t T = 8 S T = 8M* =

MT. Hence, the first statement in the theorem holds. Let M - I a M = u A D 1 , where D I ~ g r ( t ) . As f ; 'u fA=uA,

M - ' u M = f ; ' u f A D I . If r + q - 1 or if r l q - l and 9r (8 )c9 , D I E y ( 8 ) .9 implying f, D I = D I f,. Therefore, M - luM =

f;luDlf,.Let T=Mf;l.Therefore,T-luT=uD,.Let T = P D , where D is diagonal, and P is a permutation. Then D - I P - ' u P D = uDI. Therefore, P-luP = D U D ~ D - ' . As P-IuP is a permutation and D u D , D - ' is a monomial with the same cycles as U , D U D , D-I must be U . Therefore, P-IuP = a and u - ~ D u D - ' = D ; ~ E ~ ~ ( ~ ) . Therefore, DE^*, and P permutes the r-cycles of U among themselves and the fixed points among themselves. Hence, there are S E Z,* and V E Z;

E Gal(F,). Then, M * u M * - l = M T U T - I M - ' = M U M - ' .

such that U = P S - I V - ' fixes the r-cycles and fixed points of U .

Howcver, S, V , and P commute with a. Therefore, so does U , and therefore, ( ( s r + l)U,(sr +2)U; . .,(s + 1)rU) = U , for 0 I s I c - I . Thus, for 0 I s I c - I, there exists p 5 such that ( s r + 1 + i)U = sr + 1 +(i + P , ) ~ for 0 I i I r - 1. Hence, U = u[ ' l . . .

U~P: I E W. Therefore, M = Tf, = UVSDf, E T ' Z j Z T 2 *F. 0

In the next section, we examine cases where 8$8) = { I ) .

Lemma 5: If either g r ( 8 ) = ( I } or if Pr(8) consists only of scalar matrices and U has a fixed point, then 9* = 8.

Proof: If DEL?*, u - l D u D - ' is an element of g r ( C ) with 1 on any fixed point. Under either assumption, u-IDuD-I

0

When L2* = 2, the matrices in I V have a particularly nice

must be I . and hence, D E 2.

action on t as described in the next result.

Theorem 3: Suppose t and 8' have the same automorphism U . Suppose 8 = C(u)@E,(u)Cd . . . @E,(u) and 8' = C ' ( U ) @ E ; ( u ) $ . . . @ E ; ( u ) are the decompositions of C and 6' as in Section 11. Let M E 9'Z;Z:g.Y- and T E Gal(F,) such that C' = MT. Then, C ( U ) M T = C ' ( u ) , and E,(u)MT = El( , , (u) , where p is some permutation of 1; . ., g.

Proof If v E C ( u ) , VMTU = VU'MT = VMT as MT normal- izcs ( a ) . Hence, C ( U ) M T = C"(u). Suppose v E E,(@). If T E YXjZ?LZ , then vTI, E I , for 1 I j I c and vTl f l , = 0 if c + 11

j I c + f by looking ai the action of Y, Z;, Z:, and 9 and by using the fact that I , is an ideal of R. Notice f , E F and T E Gal(F,), when restricted to R , permute the roots of m,(X); . . ,m,(X) of Section I1 and fix l + X + . . . + X ' - ' . Therefore, f,, T fix the field I,, and permute I,; . ., Ig. If M = TfA and p is the permutation determined by (I,)fA-r = IF(,,, vTfAT1cl, E I+(,, for 1 I j I c and v T T ~ , ~ , , = 0 if c + 1 5 j I c + f. 0

The isomorphism between I,, = ( a ( l + X + . . . + X ' - ' ) ( a E

F,) and Fy is given by a(l + X + . . . + X r - ' ) + ar (see Lemma 1 of [7]). Hence, we define

I

a , ( l + x + ' . . + x r - I ) , u , . + 1 , ' . E C ( u ) } .

We now describe the natural actions that W, Z j , Z?, 8, 9-, and Gal(F,) induce on C(u)@ and E,(u>*. First, Y and F act trivially on C(u)@; Z j and C: permute the fixed points and cycles, respectively, of C ( U ) @ ; 8, being constant on cycles, multiplies coordinates of C ( U ) @ and EI(u)*; and Gal(F,) acts normally on entries in C ( u ) @ . On E,(u)*, C j acts trivially, and 2: permutes the coordinates; elements of Y multiply coordinates of E,(u)* by powers of v,, where U, IS an elemcnt of I , of order r ; elements of 9 and Gal(F,) induce permutations of the fields I,, as in the previous proof. We denote these actions of T E Y'xjZ,*LZFGal(F,) on C(u)@ and E,(u)* by T . Note that the map T --f T i;s one-to-one and that if T E

X;ZF9Gal(Fq), TI,,,, --* TJ,,,,, is one-to-one. We have the following useful Corollary.

Corollaty I : Suppose 8 and G' have the same automorphism u, and C(u)= C ' ( u ! . Suppose C T = C ' , where T E Zjz,*9Gal(F,). Then, T is an automorphism of C(u)@. 0

IEEE TRANSACTIONS ON I N F O R M A T I O N TIIEOKY. VOI.. 36. NO. 3. MAY 1 9 9 0 655

I v . APPLICA rIONS TO QUATERNAKY CODES

In this section, we find the extremal quaternary self-dual codes of length n = 18, 20, 22, 26, and 28, which have monomial automorphisms of prime order r 2 5. By Theorem 2 of [3], if 6' has a monomial automorphism M of order r 2 5, there is a code equivalent to t with a permutation automorphism U of order r with the same cycle structure as M . (This is not the case if r = 3; see [9].) In this section, we also give some information about the structure of permutation automorphisms of order 3. The elements of F4 will be denoted 0,1,0,0. The element a,, + a l X + . . . + a,- , X r - ' will be denoted a O a l . . . a , - , , and a + aX + . . . + ax'-' will be a.

We first describe the fields I1; . ., Ig for the values of r I 17 that we will need in this section. In all cases except r = 17, the value of g is 2. The relevant data is in Tables I and 11. The generators of I: = I , - (0}, If = I , - (0} will be denoted a, p , respectively. The inner product given is that of (2), and T , , , ~ ~ - > , ~ , is as in Lemma 2 and Theorem 1. (See Example 3 of Section 11.) The column "order" is the order of I , for 1 I i I g. In Table 11, I f , I t are generated by y , 6 , respectively. Except in the cases r = 3 and 11, p = T ~ , " - ~ , , ( ( Y ) ; if r = 17, 6 = T ~ , ~ ( Y ) as well. Note that a generator raised to the power ( 1 I , I - l ) / r generates the subgroup corresponding to the cycle shifts; multiplication by these elements to each cycle corresponds to the application of elements {TIT E W } to E,(u)*.

TABLE I 3 s r s 1 3

r ff P (u,v) Order L

3 W l O W O 1 c u,v: 7 1 . 1 4 , = I C

5 1 W O W l 1 0 0 W l c U I V P 72.1 16 r = l

examine that case further. In the tables, 6 is the group {flT E Z j Z p 3 GaI(F,)}nAut(C(a)@). (See Corollary 1 of Section 111.) The element T E Gal(F4) is conjugation. The matrices are generator matrices for the given codes, with cycle coordinates on the left. The self-dual quaternary codes of length less than 14 used in the tables and the proof are found in [12]. If x E 8, w t ( x ) is the weight of x as a vector in Fi'. If x E E,(u), let x * E E,(u)* be associated with x as in Section I1 and W t ( x * ) be the weight of x * viewed as a c-tuple in I,'. If x@ EC(U)@, W t ( x @ ) will be the weight of x@ as a vector in F4(.+f'.

Theorem 4: Let 8 be an extremal [ n , n / 2 ] self-dual code over F4 with a permutation automorphism (T of prime order r 2 3 with c r-cycles and f fixed points. For 18 I n I 28, n # 24, the only possibilities are as follows.

1) n = 18, r = 17, c = f = 1 (see Table 111). 2) n = 18, r = 5 , c = f = 3 (see Table IV). 3) n = 18, r = 3, c = 6, f = 0. 4) n = 20, r = 5 , c = 4, f = 0 (see Table V). 5 ) n = 2 0 , r = 3 , c = 6 , f = 2 . 6) 12 = 22, r = 11, c = 2, f = 0 (see Table VI). 7) n = 22, r = 7, c = 3, f = 1 (see Table VII). 8) n = 22, r = 5 , c = 4, f = 2 (see Table VIII). 9) n = 22, r = 3, c = 6, f = 4.

10) n = 26, r = 13, c = 2, f = 0 (see Table 1x1. 11) n = 26, r = 5 , c = 5, f = 1 (see Table X). 12) n = 26, r = 3, c = 8, f = 2. 13) n = 28, r = 13, c = f = 2 (see Table XI). 14) n = 28, r = 7, c = 4, f = 0 (see Table XII).

Proof: We examine the case n = 28 in detail. The other cases are similar. Before proceeding we need the following two results; the first is due to Conway and Pless [2].

Lemma 6 (see Theorem 7 of 121): If 8 has minimum distance d , then

13 l l O W 0 0 ~ W 0 ~ 0 O O 1 1 0 ~ 0 0 ~ ~ 0 0 0 W ~ u,v? T ~ . ~ 4096 Proof: As 8 is self-dual, any d - 1 coordinates of t are independent. As t = C ( a ) @ E ( c r ) and E ( u ) is 0 on the fixed points, any d - 1 or fewer fixed point coordinates of C(U) are independent. Hence, dim C(U) 2 min(d - 1,f). As dim C ( a ) =

0

r = l

(1/2Xc + f) by Theorem 1 of [7], the result follows. TABLE I1 r = 1 7

generators (u,v) T , , , ~ , - > , ~ , Order Lemma 7: C(U)@ is self-dual.

(Y = l l W 0 1 W W W W 1 0 W 1 1 0 0 0 Proof: As C(U)@ is a [ c + f , ( c + f ) / 2 ] code and r = 0 c 1 mod2, the result holds by Theorem 1 of [7].

p = 1lwOlwEoW10wll0O0 U,";h 72.1 256 i = I

y = l W W w l 0 w W 0 1 W W w l o o o

s = l ~ W w 1 0 w W 0 l W W w l 0 0 0

The next theorem gives basic information about the codes we are examining. The codes are self-dual over F4 under the inner product ( l ) , where m = 2; i.e., ( u , v ) = E:'= ,u,vf. The left and right duals are the same for inner products (1) and (2); therefore, left and right can be omitted when applying the results of Section 11. The codes we are examining are extremal with minimum distance d = 2[n/6]+2; that is, d = 8, for n =

18,20,22, and d = 10 for n = 26,28. In the statement of the theorem, the codes are described by the data in the tables. We do not include such tables for the ease r = 3 because wc do not

TABLE IV n = 18, r = 5, c = f = 3

C(U)Q, E,(U)* E,(u)*

[ p'" ""1 l O 0 l w [ o 1 0 1 1 :] [d) ,(I

O O l l W l 0 p" p"

656 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 36, NO. 3, MAY 1990

TABLE V n = 20, r = 5, c = 4, f = 0

Form E , ( o ) * EAU )* Restrictions

01152 01k52 0 5 1 5 1 4

A = p4t1+1)+ P 4k I f k - i mod 15

6 = ((1,2,3,4),(1,3), diag{w,l,w,l),~); 161= 144

TABLE VI TABLE VI1 n = 22, r = 7, c = 3, f = 1 n = 22, r = 11, c = 2, f = 0

C b ) @ E ~ ( P ) * E,(u)* Restrictions C(U)@ E l ( a ) * E,(u)* Restrictions

TABLE VI11 n = 22, r = 5, c = 4, f = 2

Form E l ( a ) * E>(u)* Restrictions

I I1 [ad) :k 4:] [ 0 p" p - 4 k 0 po 0 0 p - 4 ' O s j < k < 2

0 1 j r 2 01k12

po p - 4 k 0 0 1 / 5 2

TABLE IX n = 26, r = 13, c = 2, f = 0

C(uM E ~ ( u ) * &(a)* Restrictions

IEEE TRANSACTIONS ON INFORMATION THEORY, VOI.. 36, NO. 3, M A Y 1990 657

TABLE X n = 26, r = 5 , c = 5, f = 1

Restrictions

0 5 i 5 2 O i j 5 2 O 5 k 5 2 0 1 1 1 2

0 1 r n 5 1 4 O 1 n 5 1 4

j + n $ k + r n mod15 i + n P l + k mod15 i + rn + I + jmod 15

TABLE XI n = 28, r = 13, c = f = 2

C(u )@ E ~ ( u ) * E2(u)* Restrictions

We proceed with the proof of the case n = 28 of Theorem 4. If r 2 17, c = 1, and f~ 11; because d = 10, this is not possible by Lemma 6. Lemma 6 eliminates r = 13, c = 1, and f = 15. Therefore, c = f = 2 is the only possibility when r = 13. As C ( a ) @ is a [4,2] self-dual code, it is C,eC, in the notation of [12]. Therefore, C ( u ) @ is as in Table XI. As I , contains elements of weight 8 (the generator a for example), the code E,(a)* cannot have a generator matrix with 0 in either column. Therefore, EI(u )* is generated by [a",a']. We scale columns 2 and 4 by elements from F4 and cycle column 2 (i.e., apply powers of a40'5/13 to column 2, which has no effect on C(a)@); we may assume 0 I i < 4095/39 = 105. By reversing columns 1 and 2 (and 3 and 41, we get the generator matrix [aO,ap ' ] . Multiplying column 2 by a1O5 (and appropriately scaling column 4), we may assume 0 I i < 52. By Theorem 1, E2(a)* is the dual

under the inner product of Table I of T , , ~ ( E , ( ~ ) * ) , giving the generator matrix of Table XI.

If r = 11, T ~ , , ~ - \ , ~ = T ) , ~ , which is the identity. Hence, El (a )* is self-dual, and c is even by Theorem 1. Therefore, c = 2 and f = 5 , contradicting Lemma 6.

If r = 7, then c = 1 is impossible as E J a ) or E,(a) would have a nonzero vector of weight less than 7. Cases c = 2 and 3 are eliminated by Lemma 6. Therefore, if r = 7, c = 4 , and f= 0. Hence, we may assume C ( a ) @ is as in Table XII. The map f 3 interchanges I , and I,. We may therefore assume dim,, EI(a)* I 2. Suppose EI(u )* is generated by [a", a', a', a.]. By cycling columns 2, 3, and 4 (multiplying by powers of a'), we may assume 0 I i, j , k I 8. By multiplying by w or W in columns 3 and 4 and then recycling, we may assume 0 I k I 2. Inter- changing columns 1 and 4 and columns 2 and 3 (and recycling and rescaling), we may assume 0 I k I 1. E2(u)* is the dual of T ~ , - , (El(a)*) by Theorem 1. Therefore, E,(a)* is generated by

[ :;":.I. (6) P" p8IpXk

By examining I,, wt(P") = 4 if 31x. Therefore, if k = 0, row 1 of (6) has weight 8, which is a contradiction. Therefore, k = 1. Similarly 3 + 8i - 8, 3 + 8 j - 8. Therefore, i # 1,4,7 and j # 1,4,7.

TABLE XI1 n = 28, r = 7 , c = 4, f = O

Form EAU)* Restrictions

658 IEFE TRANSACTIONS O N INFORMATION T l i F O R Y . VOL. 36. NO. 3, M A Y 1990

We also obtain vectors (px',O,p",O) and (px',p",O,O). Hence, 3 1 8 j and 3 + 8 i . Therefore, i # 0 , 3 , 6 and j # 0 , 3 , 6 . Hence i, j E {2,5,8}. We obtain (O,p", px'-xj,O), and therefore, 3 + Xi - S j . Thus, i and j cannot both be in (2,5,8). Therefore,

TABLE XI11 THE ACTION OF YANV T

Generator r of 9- Action of generator of ,Y Action of T

dim,, E,(a)* = 2. Suppose we have a vector v* E E, (a )* , where 5 f2 f 2 ( c U ) = p 2 , f 2 ( p ) = a 2 C Y T = p , p T = a

If i E {1,2,4,5,7,8}, wt(1100+ v ) = 8. Similarly there is no v* = 11 f 2 fi(a)=p:, f i ( p ) = a G ' T = p , p T = a 13 f 2 f2(Q')=p , f2(p)=(u2 a T = p , p T = a

v* = (a", a',O,O). By cycling, we assume 0 5 i I 8. If 3(i, wt(v) = 8.

of E,(a)* to

7 f3 f 3 ( a ) = P 4 , f 3 (p )=a ; a7=ax, P . = P X

(0,0, a'), a'). Therefore, we can row reduce the generator matrix

( 7 )

If det(: :)=0, we get a vector v* as above. If a = d = 0, b = a', c = ak, then E,(u)* is generated by

However, wt(a" + p") = 2, and therefore, 2 I wt((aO, 0,0, a')+ (p" ,0 ,0 , /33h-X'))~ 9. By interchanging columns 3 and 4 and rows 1 and 2 of (71, we get Form I or Form I1 of Table XII. The bounds on the exponents are obtained by cycling and scaling appropriately columns 2, 3, and 4 as was done previously. E,(o)* is the dual of T,, - , (E,(a)*).

If r = 5 and c = 1 or 2, E,(u)* contains a nonzero vector of weight less than 10 for some i. Lemma 6 eliminates c = 3 or 4. Therefore, if r = 5, c = 5, and f= 3. By Lemma 7 , C(a)@ is an [8,4] self-dual code. By [12], C ( u ) @ = C,@C,@C,@C,, C,@E,, or E,. Clearly, there cannot be x E C(a) such that W t ( x @ ) = 2 and x@ nonzero on some fixed point. Hence, C(a)@ # C,@C, @C,@C, and if C(u)@=C,@E,, the coordinates of the subcode E, consist of three fixed points and three cycles. There is, however, a vector x @ in the E, component with Wt(x@)=4 and nonzero on the three fixed points as E, is an MDS [6,3,4] code; but then, w t ( x ) = 8. Therefore, C(u)@ = E,; but as the automorphism group of E, is triply transitive, there is a vector x @ with W t ( x @ ) = 4 and nonzero on the three fixed points, which is a contradiction. Therefore, r = 5 is impossible.

Suppose r = 3. As T,,,,~->,~, = T,,, is the identity, E,(o)* are self-dual, and c is even. Note that I, and I, are F4. If the minimum distance of E,(u)* is 2, there is a vector x* E E , ( o ) with Wt(x*) = 2, and hence, w t ( x ) = 6. This implies by [12] that c 2 6. The case c = 6 is out by Lemma 6. Therefore, c = 8, and f = 4. Clearly, C ( a ) @ cannot have a x @ with Wt(x@) = 2, and if x@ E C(cr)@ with W t ( x @ ) = 4, at least three of the coordinates of the support of x@ must be cycle coordinates. Hence by [12], C ( u ) @ = E,@E,, E,,, C,,, D,,, or F,,, but an easy examina- tion of the vectors in these codes of weight 4 shows that in the above list, at most two, one, two, two, or three coordinates, respectively, could be fixed points. This contradicts f = 4.

The arguments for the other cases of n follow in similar ways. The list in the theorem consists of all forms that were not eliminated by hand. 0

In Tables 111-XII, the groups d are found from information in [12] using Corollary 1 .

By Theorem 4, Lemma 4, Theorem 2, and Lemma 5, if t and 8' are equivalent extremal self-dual codes of length n E (18,20,22,26,28} with a monomial automorphism of order r 2 5 (which we may assume is a permutation by Theorem 2 of [2]), then t'= b T where T E WZjXz,*9FGal(F4) , except for the case n = 26, r = 5. As we will see later, in the case n = 26, r = 5, no extremal code exists, and the issue of equivalence becomes

irrelevant. It is easy to see that WX.jX,*99-Gal(F4)= X.jXz,*9GaI(F4)FP. For each of the possible outcomes in Theorem 4, where r 2 5, we tested for equivalence as follows: Let 8 be a code from one of the tables of a given form. If 8" is equivalent to 8, C ( a ) = C ' ( u ) by the tables of Theorem 4. As 8' = &TS, where T E CjCz,*L3 Gal( F4) and S E 97, by Corol- lary 1, f E 6, where G is ,as in the table. We applied (by computer) each element of G followed by elements of 9 (see Table XIII) and then shifted the columns cyclically (applied elements of W ) to get a code in the range of the table until we tested all possible group elements or produced a previously tested code. We remark that because of Theorem 3, the transformations always send a code of a given form to a code of the same form because in each case, the transformations all map cycles to cycles, and the form is determined by the weight distribution of the elements of E,(u)* . Therefore, codes of different forms are inequivalent.

If t was not equivalent to a previous case, then we looked for low weight vectors. In the cases r 2 11, this was done by check- ing all possibilities up to cyclic shifts and scalar multiples. For r = 5 and 7, this was done more efficiently as follows: Represen- tatives of all one-dimensional subspaces of each code C ( a ) @ , E,(u)*, and E,(u)* were stored in lists by their weight (Wt) in F{+f, I,, and I,, respectively. For concreteness, suppose we are in case n = 28 and r = 7. Suppose there exists a nonzero vector x ~ b with w t ( x ) s X . Then as x is nonzero on at least two cycles, wt(xl , , ) r4 for some i. A list is kept of generator exponents of all elements of lo@l,@12 such that the associated ring element has some fixed weight t up to shifts and scalar multiples where 1 I t I 4. By starting from t = 4 and going to f = 1, we combined all possible vectors from C(o)@, E,(a)*, and E2(u)* each of weight at most [8/ t ] using the representa- tives of the one-dimensional subspaces such that some cycle had weight t . If a low weight vector exists, this process will find one. It proved to be quite efficient.

Table XIV summarizes the computer calculations. Because Theorem 2 may not be valid if n = 26 and r = 5, the number of inequivalent codes may be smaller than indicated in the table. The next five theorems describe the extremal codes that were obtained.

Theorem 5: Consider all [ 18,9,8] self-dual codes over F4 with a monomial automorphism of prime order r 1.5. Then, the automorphism can be assumed to be a permutation U of order r , and all such codes are equivalent to one of the following.

659 IF.EF TKANSAC'TIONb O N I N F O K M A T I O N THEORY, VOI . 36, N O . 3. MAY 1990

TABLE XIV COMPUTER RESULTS

2 ) r = 7 , c = 3 , f = 1 , and g e n C ( c r ) @ = [ i i I;], Number of Number of Number of

Table Codes Inequivalent Extrema1 n r Number Form Tested Codes Codes

18 17 111 1 1 18 5 1v 1 1 20 5 v 1 9 4 20 5 V I1 126 10 22 11 VI 66 10 22 7 VI1 27 6 22 5 VI11 I 18 4 22 5 VI11 I1 6 3 22 5 VI11 111 27 6 22 5 VI11 IV 378 10 26 13 IX 158 20 26 5 X 14742 83 * 28 13 XI 53 9 28 7 XI1 I 54 11 28 7 XI1 I1 1.5066 295

*Actual number of equivalence classes may be smaller.

1 1 1 1 3 2 2 0 3 4 0 0 0 0 3

gen E 2 ( m ) * = [ :: E::], where (YO = OWwwW, and

p" = o w w w w .

The codes in 1) and 2) are equivalent.

Proofi The fact that the codes in I) and 2) are extremal was verified by computer. The two codes are equivalent because they must be the code SI, of [12]; this code has automorphism group of order 48960 and is thoroughly discussed in [l].

Theorem 6: Consider all [20,10,8] self-dual codes over F4 with a monomial automorphism of prime order r 2 5. Then, the automorphism can be assumed to be a permutation (T of order r = 5 with c = 4 and f = 0. All such codes are equivalent to one of the following two 8=C(cr )@El(o)@ Ez(a ) , where genC(a)@

= 6 < ~ . pll

Here, cy0 = OWwwW, crl = I w O w l , ah = wWOWw, P o = OwWWw, 0 p4 = I l W O W , py = WWwOo, and PI4 = w w l O l .

Theorem 7: Consider all [22,11,8] self-dual codes over F4 with a monomial automorphism of prime order r 2 5 . Then, the automorphism can be assumed to be a permutation U of order r . All such codes are equivalent to one of the following 6= C(U) E, (a )@E, (a ) , where

I) r = l l , c = 2 , f = O , and g e n C ( a ) @ = [ l I], g e n E , ( u ) * = [a" a3"], gen E,(U)" = [ P o p"'] with the following three possibilities: i = 0 , j = 3 or i = O , j = 11 or i = 1, j = 2. These three codes are inequivalent. Here, a') = lwWwwwWWOw0,

w ~ ~ l l l ~ l L i i ~ 1 , py3 = w~WWWWWWWOW, and p'" =

~ w l w w w l l l w l .

a31 - - o w l w W l w l l l w , p" = l W w O W W w w w W w , p h 2 =

gen EI((T)* = [ a ( ! a' all, gen ~ , ( o ) * = [ :: " '-"] with the

following two possibilities: i = 0, j = 3, or i = 1, I = 3. These two codes are inequivalent. Here, a''= 1001011, a' = IWWwOlw, a-i = 0 ~ 0 w ~ ~ 0 , p O = 1110100, p - I h = 00lWWw1, and p-24 =

w o w 000 w .

6" ?-

3 ) ~ = 5 , ~ = 4 , f = 2 , a n d g e n C ( ( ~ ) @ =

[all 0 p4ll-Al P O ] 0 p-4A 0 with the following three possibilities:

= [: 2: with the following four possibilities:

k = 0, 1 = 2 ( A = l w w l O ) , or k = 0, 1 = 5 ( A = O W l l W ) , or k = I, 1 = 0 ( A = IWOOl), or k = I, 1 = 6 ( A = 0 ~ 1 0 1 ) .

Here a' = OWwwW, a' = I w O o l , a' = I W W l O , a5 = O l W W l , ah

W l l W 0 , p8 = IOlww, pY = OWWOW, and p" = w l 0 l w . The nine codes in a), b), and c) are inequivalent.

over F4 with a monomial automorphism of prime order r 2 5.

Theorem 9: Consider all [28,14,10] self-dual codes over F4 with either a monomial automorphism of prime order r 2 5 or a permutation automorphism of order r = 3. Then, the automorphism can be assumed to be a permutation U of order r = 7 with c = 4 and f = 0. All such codes are equivalent to one of the following three inequivalent codes: C = C ( ( T ) @ E ~ ( U ) @ E ~ ( ~ ) ,

= w w o o w , p" = o w w o w , p4 = I l W O W , p5 = O l w w l , p7 =

Theorem 8: There does not exist a [26,13,10] self-dual code

gen E,(u)* = [ A 0 A B n J 6" ",] with either i = 0, j = I, k = I, 1 = 9

( A = 0WwwlWl) or i = 0, j = 1, k = 1, I = 43 ( A = WWOWOOW) or i = 1, j = 4, k = 4, 1 = 28 ( A = wOw0Wll) . Here, a') = 1001011, a1 = l W W w O l w , a4 = IWOwWwl, a y = 1100101, aZX =

OlwlWWw, a 4 3 = Wwwl0Wl, p"= 1110100, p8= lWlOWww, py = 1101001, pZy = wlwOlWW, p32 = IlWwWOw, and p35 = 0Wwwl Wl.

CONCLUSION

We conclude with a few remarks. First, the cases where r = 3 of Theorem 4 were not considered further because it is more difficult to decide equivalence between two codes because the hypothesis of Theorem 2 may not hold. The author has com- pleted this case and plans to publish it in the future. Second, it would be interesting to decide on the equivalence or inequiva- lcnce of two extremal codes of length n constructed from two

660 IFFC TRANSACTION5 ON INFORMATION THFCIRY. VOL. 36, N O . 3, MAY 1990

different values of r . In the case n = 18, the codes of Theorem 5 are equivalent because the automorphism group is known (see [l]). The two extremal codes of length 20 of Theorem 6 and the three of length 28 of Theorem 9 are inequivalent. However, in Theorem 7, the only information known about equivalence of these length 22 codes is as stated in the theorem. Third, it is still unknown if there exists an extremal self-dual code of length 26 (see [5 ] ) . Finally, the programming required for this correspondence was done on an AT&T 6300 in Pascal.

131

151

161

REFERENCES Y. Cheng and N. J. A. Sloane, “The automorphism group of an [18,9,81 quaternary code,” Discrete Math., to be published. 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. J. H. Conway and V. Pless, “Monomials of orders 7 and 11 cannot be in the group of a (24,12,10) self-dual quaternary code,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 137-140, 1983. 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. Inform. Theory, vol. IT-25, pp. 312-322, 1979. J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices, and Groups. New York: Springer-Verlag, 1988. W. C. Huffman, “Automorphisms of codes with applications to extremal doubly even codes of length 48,” IEEE Trans. Infonn. Theory,

W. C. Huffman, “On the [24,12,10] quaternary code and binary codes with an automorphism having two cycles,” IEEE Trans. Inform. Theory, vol. 34, pp. 486-493, 1988. W. C. Huffman, “On the equivalence of codes and codes with an automorphism having two cycles,” Discrete Math., to be published. W. C. Huffman, “On 3-elements in monomial automorphism groups of quaternary codes,” IEEE Trans. Inform. Theory, this issue, pp. 660-664. W. C. Huffman and V. Y. Yorgov, “A [72,36,161 doubly even code does not have an automorphism of order 11,” IEEE Trans. In,forrn. Theory, vol. IT-33, pp. 749-752, 1987. C. W. H. Lam and V. Pless, “There is no (24,12,10) self-dual quaternary code,” to be published, in IEEE Trans. Inform. Theory. F. J . MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, “Self-dual codes over GF(4),” J . Cornbin. Theory, vol. A 25, pp. 288-318, 1978. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes. New York: North-Holland, 1977. V. Pless, introduction to the Theory of Error-correcting Codes. New York: Wiley, 1982. V. Y. Yorgov, “Binary self-dual codes with automorphisms of odd order,” Proh. Infonn. Trans., vol. XIX, pp. 11-24, 1983. V. Y. Yorgov, “A method for constructing inequivalent self-dual codes with applications to length 56,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 77-82, 1987.

Vol. IT-28, pp. 511-521, 1982.

On 3-Elements in Monomial Automorphism Groups of Quaternary Codes

W. CARY HUFFMAN

Abstract -A general theory is given for examining quaternary self-dual codes having monomial automorphisms of order a power of 3. This theory extends similar work when the codes have nontrivial odd order permutation automorphisms.

I. INTRODUCTION

We develop a theory for examining quaternary self-dual codes that have a monomial automorphism of order a power of

Manuscript received February 8, 1989; revised October 3, 1989. This work was presented in part at the AMS Regional Meeting, in the special session on “Codes and Designs,” Chicago, IL, May 1Y-20, 1989.

The author is with the Department of Mathematical Sciences, Loyola University of Chicago, Chicago, 1L 60626.

IEEE Log Number XY33622.

001 8-9448/90/0500-06(

3. If such a code has a monomial automorphism of prime order p > 3, by rescaling the basis vectors, the code is equivalent to a self-dual code with a permutation automorphism with the same cycle structure as the monomial automorphism (see [2]). This is not true when the monomial automorphism is of order a power of 3. A theory of decomposing such codes is given in Section 11.

In Section 111 we describe applications of this theory. General references for the terminology of coding theory are [SI and [9].

11. MAIN RtsuLTs

Let 6, bc the finite field of order q. Let P be a linear code over F,, of length n and dimension k . The weight of a vector x E 6;‘ is the number of nonzero entries in x. The minimum distance d of 8‘ is the minimum nonzero weight of all the vectors in B. C is called an [n, k ] or an [ n , k , d ] code.

Let ,A,, be the set of monomial matrices with entries from F4 = (0,1, w , U’}; an clement of M E ~A,, can be decomposed as M = PD where P is a permutation matrix and D is a diagonal matrix. The identity of k,, will be denoted by I. Let T denote conjugation in F4 and A,,* = 4, U { M T ( M E ~A,,}; here if v E F i , VMT is computed by applying M and then conjugating. Let A u t ( & ) = (T E &,,*lPT = 8‘) be the automorphism group of 6. t is equivalent to 8’ if CT = 8‘ for some T E A,,*.

Lemmu 1: Let c? be an [ n , k , d ] code over F4 with d > l . Suppose for any distinct pair of Coordinates a , b there exist vectors v 1 , v 2 ; . .,v, E 8 of weight d with a in the support of v l , b in the support of v,, such that the supports of v, and v , , , overlap for 1 I i I t - 1. Then the diagonal elements of Aut (&) are in ( w l ) .

Proof: Let D = diag(d,;. . ,d,,)E Aut (B) . The result is clear by the hypothesis if given coordinates a,P in the support of a vector x = (xI; . .,x,,) E k of weight d , we show d, = d,. But y = dUx - X D is of weight less than d and hence must be 0.

Lemma 2: Assume C is an [ n , k , d ] code over F4 such that the diagonal elements of A u t ( 6 ) are ( w l ) . Suppose A u t ( k ) contains an element M = PD where P has c 2 1 3-cycles and f fixed points with n = 3c + f and D = diag(d,; . ., d,,). Then 8 is equivalent to a code having the automorphism in Fig. 1 or Fig. 2.

Proof: Order the coordinates so that P = (1,2,3) . . . (3c - 2,3c - 1,3c). Denote the basis by PI;. . ,P I , . For 1 I i I e,

Thus d, = d,. 0

P31-2M = d3 , - IP3,- I ) P 3 , - IM = d, ,P , , , and P 3 l M =

4 1 - Z P 3 1 - 2 . Let P ~ l - l = ~ 3 1 - l P 3 c - l and P.\,=d3,-Id3;P3l~ Re-

= Pi , - 1, P3 ~ ,M = Ps,, and P3,M = a;P3/-2

placing P 3 / - by l,P.(,, which sends k to an equivalent code, and then dropping the primes, we may assume that for 1 I i I e, ( w h e r e a , = d 3 r - 2 d 3 1 - , d 3 , ) . T h e n M =

d i a g ( a , , a , , a , ; . . , ~ , , a , , a , . , d : : , . + ~ , . . . ,d::, .+,)~ A u t ( 6 ) . But d i c + l = . . . =d:,+,:=l as d 3 c + r ~ { 1 , ~ , ~ 2 } . Hence, by our assumption that the diagonal elements of Aut(#) are ( w l ) , there are two possibilities: f > 0, which implies aI = . . . = a< = d:c+ =1, and f = O , which implies a , = . . . = a , . If f > 0 or if f = 0 and aI = . . . = a< = 1, we have the automorphism in Fig. 1. If f = 0 and a I = . . . = a , # 1, we may assume, by conjugating if necessary both the code and the automorphism, that a I = . . . =

0 a< = w giving the automorphism of Fig. 2.

50$01.00 01990 IEEE

on extremal self-dual quaternary codes of lengths 18 to 28. i

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