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

1206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

0018-9448/91/0700-1206$01 .OO 01991 IEEE

by our assumption about x , we must have a = b, and d(x,C,) I d ( x , y ) I 1 as claimed. This completes the proof of Theorem 4.

0

Theorem 5: If there is a (3,O)-subnormal (3,r;M)R code then for all n 2 2

K,(n + r - 1 , R + 1) I K,(n, 1)M/3.

Proof: This follows immediately from Corollary 1, Theorem 4, and Remarks 1-3 (for n = 2 the statement is trivial; n 2 3 implies that the sets Ci in Theorem 4 are all nonempty).

Remark 7: The direct sum of a (3,n,, M,)R code and a (3, n2 , M2)1 code is a (3, n , + n2, M,M,)R + 1 code. By using the previous theorem and a (3,O)-subnormal (3,n1 + l,Mfl)R code we get a (3,n, + n 2 , M)R + 1 code, where M I M o M , /3. By Example 3 we can always choose M,, = 3M; and we can often find a smaller MO thus improving on the direct sum.

We have not been able to extend Theorem 5 to the general q. However, we can prove the following result, cf. [ l l , Theorem 11.

Theorem 6: If n 2 2 and is a ( q , n, M)1 code with minimum distance at least 2, then the once punctured code C has (q,l)-subnorm 2.

Proof: We assume that I? is a ( q , n , M ) l code 5 t h minimum distance at least 2 and denote C, ={c l ( c , i )E C}. These sets are disjoint because the minimum distance of is at least 2. Suppose that x is naughty, i.e., x E C, but d(x ,C , )? 3 for some i , j , i f j . W.1.o.g. j = O . For every y E B , ( x ) we have ( ~ , O ) E B,(I?) and y belongs to exactly one of the sets C , , C 2 , . . . , C q - - l . The points x , x t ( l , O , O ; . . , O ) , x + (2,0,0,. . . , O), . . . , x + ( q - 1,0,0,. . . , 0) are all distance 1 apart from each other and one of the sets C,,C,;. .,C,-] contains at least two of them, contradicting the fact that C has minimum distance at least 2. Hence there are no naughty points for c.

The same argument also shows that all the sets C, are nonempty. The rest of the proof is exactly the same as in the proof of Theorem 4. 0

ACKNOWLEDGMENT

The author would like to thank P. OstergHrd for sending a preprint of [14] and the referees for useful comments.

REFERENCES

[l] W. Chen and I. S. Honkala, “Lower bounds for q-ary covering codes,” IEEE Trans. Inform. Theory, vol. 36, no. 3, pp. 664-671, May 1990.

[2] G. D. Cohen, A. C. Lobstein, and N. J. A. Sloane, “Further results on the covering radius of codes,” IEEE Trans. Inform. Theory, vol. IT-32, no. 5, pp. 680-694, Sept. 1986.

[3] R. L. Graham and N. J. A. Sloane, “On the covering radius of codes,” IEEE Trans. Inform. Theory, vol. IT-31, no. 3 pp. 385-401, May 1985.

[4] H. 0. Hamalainen and S. M. Rankinen, “Bounds for football pool problems and mixed covering codes,” J . Combin. Theory, A., vol. 56, no. 1, pp. 84-95, Jan. 1991.

[SI I. S. Honkala, “Lower bounds for binary covering codes,” IEEE Trans. Inform. Theory, vol. 34, no. 2, pp. 326-329, Mar. 1988.

[61 -, “Modified bounds for covering codes,” IEEE Trans. Inform. Theory, vol. 37, no. 2, pp. 372-375, Mar. 1991.

171 -, “All binary codes with covering radius one are subnormal,” Discrete Math., to appear.

t81 -, “On the normality of codes with covering radius one,” in Proc. Fourth Joint Swedish -Soviet Int. Workshop Inform. Theory, Gotland, Sweden, 1989, pp. 223-226.

[9] 1. S. Honkala and H. 0. Hamalainen, “Bounds for abnormal binary codes with covering radius one,” IEEE Trans. Inform. Theory, to appear.

[lo] A. C. Lobstein and G. J. M. van Wee, “On normal and subnormal q-ary codes,” IEEE Trans. Inform. Theory, vol. 35, no. 6, pp. 1291-1295, Nov. 1989.

[ l l ] G. J. M. van Wee, “More binary covering codes are normal,” IEEE Trans. Inform. Theory, vol. 36, no. 6, pp. 1466-1470, Nov. 1990.

[12] -, “Bounds on packings and coverings by spheres in q-ary and mixed Hamming spaces,” J. Comb. Theory, A. submitted.

[13] P. R. J. Ostergird, “A new binary code of length 10 and covering radius 1,” IEEE Trans. Inform. Theory, vol. 37, no. 1, pp. 179-180, Jan. 1991.

[14] -, “Upper bounds for q-ary covering codes,” IEEE Trans. Inform. Theory, vol. 37, no. 3, pt. I, pp. 660-664, May 1991.

On Extrema1 Self-Dual Quaternary Codes of Lengths 18 to 28, I1

W. Cary Huffman

Abstract -A general decomposition theorem is applied to find all extremal self-dual quaternary codes of lengths 18 to 28 that have a nontrivial monomial automorphism of order a power of 3. Techniques to distinguish these codes are also presented.

Index Term -Extrema1 quaternary codes, automorphisms of codes, self-dual codes.

I. INTRODUCTION

In [8] a decomposition theory of quaternary self-dual linear codes t that pos’sess a certain monomial automorphism of order 3 or 9 is developed. In Section 11, we summarize these results; we also show that, when considering extremal codes which have an automorphism of order a power of 3, we may assume the automorphism is of the types examined in [SI.

In [4] and [ll], all self-dual codes over F4 of length at most 16 are enumerated. An exhaustive search for higher lengths seems infeasible. Still one would like to enumerate the extremal codes if possible. In [lo] it is shown that there does not exist a [24, 12, 101 self-dual code over F4. (See also 121, [3], [7], and 181.) In [9], all inequivalent extremal quaternary codes of lengths 18, 20, 22, 26, and 28 that have a monomial automorphism of prime order r 2 5 are enumerated. This correspondence extends those results to include the prime r = 3.

In Section 111, we present situations in which the equivalence of the codes under consideration can be decided. In Section IV, the applications of the theory to lengths 18 to 28 are presented. General references to coding theory are [12] and [13].

11. TERMINOLOGY AND BACKGROUND RESULTS

Let F, be the finite field of order q. Let 8 be a linear code over F, of length n and dimension k. The weight of a vector x E F i , denoted wt(x), is the number of nonzero entries in x.

Manuscript received March 28, 1990; revised December 6, 1990. This work was presented in part at the SIAM Annual Meeting, Chicago, IL, July 16-20, 1990. The programming for this work was done in Pascal on an AT&T 6300.

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

IEEE Log Number 9143037.

- - 1

I

M =

1207 IEEE TRANSAmIONS ON INFORMATlON THEORY, VOL. 37, N O . 4 , JULY 1991

By Theorem 3 of [8] , 8 is a self-dual code over F4 under (1) if and only if 8* is a self-dual code over F64 under (2).

Suppose I is a quaternary code with a nontrivial automorphism (i.e., not in (ol)) of order a power of 3. If the only diagonal elements of Aut (8) are (o I ) , then by Lemma 2 of [81,

' I is equivalent to a code with an automorphism of Type I or Type 11. We show that, with one exception, extremal self-dual

Lemma I : Let 4 be an [ n , n / 2 , d ] extremal self-dual quaternary code. Then the only diagonal elements of Au t (8 ) are ( w I ) except when I is the [4 ,2,2] code with generator matrix

A 1

1 W

0 codes satisfy this criterion. o2

\

The minimum distance d of k is the minimum nonzero weight of all the vectors in 8. 8 is called an [n , k ] or an [n , k , d ] code.

Let kn be the group of all n x n monomial matrices over F4 = {O, 1 , w , w 2 } where o2 = W = 1 + o. Let Gal(F4) be the Ga- lois group of F4 over F2, and let L: be the semidirect product of Ln extended by Gal(F4). If T EJ;, we write T = PDv where P is a permutation matrix (permutation part), D is diagonal (diagonal part), and Y E Gal( F4). Linear codes k and 4' over F4 are equivalent whenever 8'= kT for some T E k:. Define the automorphism group of d as Aut (&) = {T E k:I%T = k}. Let I be the n x n identity of &:.

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

( u , c > = u p : , ( 1 ) r = l

where u = ( u l ; . . , u n ) , c = ( u 1 ; ~ ~ , u n ) are in FJ. If 8 is an [ n , k ] code over F4, define ={U E FTJ(u,c) = 0 for all U E 8). d is self-orthogonal under (1) if 8 G tf I and self-dual if B = 8 l. Note that applying elements of k: to a self- orthogonal code transforms the code to a self-orthogonal code. If e is an [ n , n / 2 , d ] self-dual code, d I 2 [ n / 6 ] + 2 by [ l l ] . When equality holds, we say the code is extremal.

We summarize the results of [8]. In that paper we examined automorphisms M of a code 8 where M was of Type I or Type 11, respectively.

1 of [8], 8= E 0 ( M ) @ E 1 ( M ) e E 2 ( M ) . Suppose U E F;, where U = (u1,,, ul, 1, u l , . . , u ~ , ~ , , uC, 1, uC,2, u3= + . . . , u ~ , + ~ ) . Associ- ate to U the element U' E RC X F[ where U'= (U;, . . . , u)c,~;~+~,...,uj~+~)with U ' , = U , , ~ + U , , ~ X + U , , ~ X ~ for l s i l c and U: = U, for 3c + 1 5 z s 3c + f. Define E,(M)* as follows. If u E E k ( M ) , associate to u an element U* = (UT; . . , u : + ~ ~ ) E Fl+fk where, for 1 5 i 5 c, U: = uTi , (X) with U: E F4, and the elements u:+~;. ., u : + ~ ~ are the f k elements from u ; , + ~ , . . . , u ; , + ~ whose subscripts are in F - k . Define Ek(M)* =

{U* E F,C+fklu E E,(M)}. Theorem 2 and Corollary 1 of [8] show that if 8 is self-dual, so are E, (M)* for 0 s k I 2 and c , fo , f l , f 2 have the same parity. This process is reversible. Take any three self-dual codes over F4 of length e + f k , call them E, (M)* , identify c coordinates with the c 3-cycles of M and f k

coordinates with Fwk, and construct E k ( M ) by reversing this process. The resulting code E o ( M ) @ E l ( M ) @ E 2 ( M ) is self-dual and has M as an automorphism.

We now summarize the results of [8] regarding Type I1 automorphisms. Assume 8 is a code with a Type I1 automorphism M. Suppose there are c copies of 8. Thus n = 3c. Let ~ = ( u ~ , , , u ~ , ~ , u ~ , ~ , ~ ~ ~ , ~ ~ , ~ , u , , ~ , ~ , , ~ ~ ~ F;. Associate U* with U where U* =(U:;. .,U:) with UT = U , , , + u , , , X + u , , ~ X ' . Each uT is in F 4 [ X ] / ( X 3 - o) = F64. Let k* = (u*(u E 8}, which is a code over FW. Define the inner product (.;I on F 4 [ X ] / ( X 3 - ~ ) by

Type Z Automorphism:

( A

with multiplicative identity i k ( X ) ; also R = Zo@Il@Z2. If U E F,", for 1 I i I c, let u I ~ . be the restriction of v to 111. EQUIVALENCE OF CODES

that we may view in the natural way as an element a, + a l X + a 2 X 2 E R. Furthermore uMln, = ( a , + a , X + a 2 X 2 ) X . Define E , ( M ) = {U E 81 vln, E Zk for 1 s i I c, U, = 0 if i E F- Y w k }

for k = 0,1,2. Notice that if U E E , ( M ) , vM = okv. By Theorem

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

First we assume k is an [ n , k ] code over F4 with an automorphism M of Type I. Let T be conjugation in F4; so Gal(F4) =


{ l ,~} . Define K = d i a g ( l , w , W , l , w , W ; . ~ , 1 , w , ~ , 1 ; . . , 1 ) , where 1;..,1 arc the values on F. Let Y , = K , Y2 = K - ' , Y 3 - -7 , Y 4 - - T K , Y , = T K - I ; let K = Y - ' w . for 1 1 i 1 5 . No- tice that &Y) is equivalent to & with N , ( w I ) ~ A u t ( & x ) . Let M I = WN,, M2 = wN2, M3 = N3. M4 = GN4, and M, = UN,. If A , is the direct sum of c copies of

4 = 0 0 1 , (: : ::I Mi = A,@diag(a,; . . , a j , b i ; . ., bi ,c i ; . . , c i ) ,

where there are fo ai's, f l hi's, and f2 ti's. Furthermore the values (a i ,b i , ci) are (0,1, w ) , (w,W, 11, ( 1 , W , w ) , (W,w, l ) , and ( w , 1, W) when i = 1 , 2, 3, 4, and 5 , respectively. This shows that if & has a Type I automorphism M , there are 5 other codes equivalent to k with Type I automorphisms where the 6 total Type I automorphisms have fixed point sizes all possible orders of {fo,fl,f2}. Thus from now on we may assume that if two of f,, f l , f 2 are equal, then f l = f2. In Section IV, we will furthermore assume whatever order among f0 , f l , f2 is convenient for the case being considered.

We define several elements and subgroups of d:. Let K be as before. Define s = (2,3), ( 5 6 ) . . . (3c - 1,3c). The effect of applying s to k is to replace X by X 2 in each 3-cycle of 4. Define U, =(3t -2,3t -1,3t) for 1 ~t I C , and W=

. . . U:< 10 I p, I 2 and 1 -E t I c}. Application of an element of W to k cycles the entries in each 3-cycle of k. If 2, is the symmetric group on {l; . .,c} and if 4 E 2,, define 4* as the permutation (3t - i)4* = 3(t4)- i for 1 I t I c, 0 I i I 2, and xd* = x for 3c + 1 I x I 3c + f. Let 2: ={+*I+ E 2,}. Applica- tion of an element of 2: to 8 permutes the 3-cycles. For 0 I i I 2, define H, as the set of all permutations of the fixed points of Fw,, fixing all other elements of 1;. .,n. Appli- cation of an element of 2 , to & is clear. Let 9= {diag(a,;. . , ~ , , ) l u ~ ~ - ~ = a3r - I = a3, for 1 s t I c). Application of an element of 9 to 8 scales each coordinate, with the scaling factor on a given 3-cycle the same on each of the 3 coordinates of the 3-cycle. If f, = f 2 , define s , , ~ by (3c + fo +

= 3c + fo + f l + i for 1 5 i I f,, (3c + f, + f l + i ) s , , z = 3c + f, + i for 1 I i I f2, and X S ~ , ~ = x, otherwise. If fo = f l = f 2 , define by (3c + i)s , , l ,2 = 3c + fo + i , (3c + fo + i ) so , l ,2 =

3c + fo + f l + i, (3c + f, + f l + i ) s , , l ,2 = 3c + i for 1 I i I f2, and XS,,,,~ = x , othenvise. Finally define 4 as follows: if f o , f l , f 2 are distinct, let / = { I } ; if fo # f l = f 2 , let 4= ( s , , ~ s ) ; if f o = f l = f 2 , let 4= ( s ~ , ~ s , s ~ , ~ , ~ K - ~ ) . Notice that ( S , , ~ S ) - ~ M ( S ~ , ~ S ) = M 2 and (S,,,,~K-~)-~M(S,,~,~K-~)= WM. Thus &'= &s, ,~s and P= &s, , , ,~K- ' have automorphism M . Furthermore, if B'= E b ( M ) @ E ' , ( M ) @ E ; ( M ) and 8"= E i ( M ) @ E ; ( M ) @ E ; ( M ) are the decompositions of B',P as in S e c t i o n 11, t h e n E b ( M ) = E o ( M ) s , , , s , E $ M ) = E 2 ( M ) ~ 1 , 2 ~ , and E ; ( M ) = E , ( M ) S : , ~ S , and E ; ( M ) =

E2(M)s , , , ,2K- ' , E ; ( M ) = E,(M)s , , , , ,K- , and E'; (M) = E l ( M ) ~ , , l , 2 K - 1 . This is clear when we note that on the cycle positions, fixes Z , and switched I , and 12, and S ( ~ , , , ~ K - ' permutes Z,, I,, Z2 cyclically.

Lemma 2: Suppose M is an .element of dfl of the form of a Type I automorphism where if two of f o , f l , f 2 are equal, fl = f2.

a) Suppose for some N E Lfl, N-IMN E ( M , w I ) .

b) Suppose for some N , E d,,, ( N,T)- 'M( N,T) E ( M , w l ) . Then N E WXfUZf1Sf22~9.Y.

Then N , E W Z f U Z f I ' C f 2 Z ~ 9 4 s . ,

Proof: Suppose first that ( N p - 'M( N17) = wiMJ. Then NC'MN, = TW~M'T-~ = w-'s-'M2's. SO ( N , s - ' ) - ' M ( N , s - ' ) E ( M , o Z ) . Letting N = N,s- ' , we have b) if a) is true.

So assume for some N E An, N-'MN = o'MJ with 0 I i I 2, 1 I j I 2. The eigenvalues of M are 1 , w , W with multiplicities c + f,, c + f l , c + f2, respectively. Because w ' M J has the same eigenvalues as M , if f,, f,, f2 are distinct, then i = 0 and j = 1, if f, f f l = f2 , then i = 0. So by definition of 9, in each case, there exists S E 9 A?,, such that S-'MS = w ' M J . Let T =

N S - ' € A f l . Hence T - ' M T = M . Let T = PD where P is the permutation part and D is the diagonal part. So P-'MP = DMD-'. Let U; E FJ have 1 in the ith coordinate and 0 else- where. Suppose i E F,x and uiP-l = U,. As M is diagonal on

w k vi P - = w k U , implying m E F u k . Thus for each 0 I k I 2, P permutes F w k . So P permutes the elements of R, U . . . U 9, = {l;. .,3c}. Let i E Rj; then uiP-'MP = u,DMD-' = a i ( u i M ) for some ai E F4. As P-'MP is a permutation on R I U . . . U R,, ai = 1 . So uiM = uiDMD-' for any i E R, U . * U R, U FW" U y"~ U F u 2 . Therefore P-'MP = DMD-' = M , and so P permutes entire 3-cycles a,,. . . , R, among themselves. In particular, D E 9 and there exists Y E 2: and V E I ; ~ ~ ~ ~ ~ ~ ~ ~ such that U = PY-IV-' fixes with 3-cycles R,; . . ,Cl , and is trivial on FwuuFwl uFw2. But P , Y , and V commute with M , and therefore U does also. Hence, ((3t -2)U,(3t - 1)U,(3t)U)= a, for 1 I t I c. Thus for 1 I t I c, there exists p I with 0 I p, I 2 such that (3t - i)U = 3t - x , where xi = (i - pl)mod3 for 0 5 i - <2; so U = u ~ l . . . u , ' " . ~ Y . S o N = T S = P D S = W Y D S prov-

F w k , WkUi=UiM=UiDMD-l=UiP-'MP=U,MP. S O U,M=

ing a). 0

Theorem 1: Let -15 and 8' have the same automorphism M of Type I. Assume ( M , wZ) is a Sylow 3-subgroup of Aut(C). Then 8 and B' are equivalent if and only if B'= &N for some N E C: c f u g ( S T ) ~ f 1 2 f 2 / ~ .

Prooj? Assume &'= &T for some T E A:. Then Aut(&') = T-'Aut(&)T. As ( M , w Z ) is also a Sylow 3-subgroup of Aut(B'), ( M , w Z ) and T( M , w1)T- l are Sylow 3-subgroups of Aut (8). So there is an S E A u t ( & ) such that ST(M,wZ)T- 'S- '= ( M , w Z ) . Let V=ST; clearly e= &V and V - ' W E ( M , w Z ) . The result follows from Lemma 2, when you observe that

If G is a group and H a subgroup, denote the normalizer in G o f H by N J H ) .

Theorem 2: Let k have an automorphism M of Type I. Then NL;((M,wZ))= 2 ~ Z f u 9 ( s T ) 2 f , 2 f 2 y W . If (M,oZ) is not a Sylow 3-subgroup of Aut(&), there exists a 3-element N E

Proof: The first statement follows from Lemma 2 and the observation at the end of the proof of Theorem 1. For the second statement, let G = Aut(&). There exists a Sylow 3-subgroup P of G containing (M,wl). As P properly contains

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991 1209

( M , w l ) , by [6], N,(( M , w l ) ) properly contains ( M , w l ) . The second statement now follows from the first. I7

We now describe the natural actions that x;, xfu, 9, ( S T ) ,

zfI, z f2 , 4, and W have on 8 and the actions they induce on E,(M)* for i = 0,1,2. Suppose 8= 8 N where 8' also has automorphism M ; let 8= E,(M)@E,(M)@E,(M) and B'= Eb(M)@E;( M ) @ E 2 ( M ) be the decompositions of Section 11. If N E x:,zfo,L3,(s~),Cf,,Cf2, or W, E : ( M ) = E,(M)N. If N = sl,,s, E@f) = E,(M)N, E' , (M) = E 2 ( M ) N , and E ; ( M ) =

E,(M)N. If N=s,,,, ,K-', EJ+,(M)= E,(M)N where the subscripts are read mod3. In particular the action of any element N E Z : z f u 9 ( ~ ~ ) C f l Z f 2 4 F ' on 8 preserves the decomposition. For the induced actions we have the following. If N E zz, N permutes the 3-cycle coordinates of each E,(M)*. If N E z,, N induces a trivial action on E,(M)* where j # i , and permutes the fixed points of E,( M ) * . If N E 9, N induces the action of a diagonal matrix on E,(M)*. If N = S T , N is conjugation on E , ( M ) * . If N = S , , ~ S , N induces a trivial action on E,(M)*, sends E,(M)* to E;(M)*, and E2(M)* to E;(M)* by interchanging wki , (X) and wki2 (X) . If N=s,,,, ,K-', N sends E,(M)* to EJ+,(M)* by sending wki , (X) to w ~ z , + , ( X ) (subscripts mod3). Finally if N = ur' . . . a[., N induces a trivial action on E,(M)*; on E,(M)* the induced action is to scale the jth 3-cycle by w F i ; on E,(M)* the induced action is to scale the jth 3-cycle by Wp,.

Let 9' be the subgroup of 5, which is trivial on R I U . . . U R, U Fwu; let 9* be the subgroup of A5, which is trivial on FwuFWz. If T E X ; Z ~ ~ ~ * ( S T ) , let T be the action of T induced on E,(M)*. The proof of the following is obvious from the previous descriptions.

Corollary I : Suppose 8 and 8 have the same Type I automorphism M and B'= 8 N with N E z:xf,,5(s~)zflxf23'Y. Then N = TQ, where T E Z : C f U 9 * ( s ~ ) and Q E Z f l Z f 2 4 9 ' Y . Furthermore E , ( M ) * f = Eb(M)*, and, in particul:r, f E

Aut(E,(M)*) if E,(M)= Eo(M). Also the map T + T , where T E Z : Z , L ~ * ( S T ) , is one-to-one.

We now discuss the case where a quaternary code has a Type I1 automorphism M. So n = 3c and ( M ) is a group of order 9; note that M 3 = OZ. Let 2: be as in the Type I case taking into account that M has no fixed points. Letting B be as in Type 11, defineW*={BFI@ . . . @ B F ' . 1 0 ~ ~ L I ~ 8 a n d l ~ t ~ c } . L e t S = (2,3)(5,6). . . (3c - 1,3c)diag(l, W, 1,1, W, 1; . ., 1, W, 1 ) ~ . We now give analogous results to Lemma 2 and Theorem 1.

Lemma 3: Let 8 have an automorphism M of Type 11. Suppose for some N E .&: that N-'MN E ( M ) . Then N E

2: w* ( S ).

Proof N- 'MN= MI for some j with 3 % j . Notice that S-'MS' = M2'. Choosing i so that 2' = j(mod9), we have N-IMN = S-'MS'. Letting T = NS-' , we have T- 'MT = M . Let 5 be a root of X 3 - w . The eigenvalues of M are 5, and . $w2= t7 each with multiplicity c. If T E A : --.U,, the eigenvalues of T- 'MT would be t2 , t8, and 16, which is impossible. So T E An. Let M = PD and T = P I D , where P , P , are permutation matrices and D , D , are diagonal. Now DP, =

P,D* and D I P = PDT for some diagonal matrices D * , DT. Then T- 'MT = M implies P-'P;'PP, = DTDD;'D*-' . As P-'P;'PP, is a permutation and DTDD,'D*+' is diagonal, both must be the identity. In particular P I commutes with P and thus P, permutes the 3-cycles of P . Thus there is a Y E 2,; such that U = Y - ' P , fixes the 3-cycles of P . As Y commutes with M , M = T- 'MT = T - ' W - ' T . But Y - ' T = U D , and

hence (UD, ) - 'M(UD, )= M . We now show U D , E Y*. AS U fixes the 3-cycles of P , we only have to show that, if W E k3 with W-'BW = B, then W = Bk for some k . But that is trivial. Hence Y - ' T = W, E W* and so N = TS' = W , S ' E x:?'*(S).

0

Theorem 3: Let 8 and 8 have the same automorphism M of Type 11. Assume ( M ) is a Sylow 3-subgroup of Aut(&). Then 8 and 8' are equivalent if and only if 8' = 8 N for some N E Z:Y'* (S) .

Proof: Analogous to the proof of Theorem 1. 0

IV. CLASSIFICATION OF EXTREMAL CODES

In this section we apply the results of the previous sections to classify the extremal self-dual quaternary [n , n /2, d] codes of lengths n = 18, 20, 22, 26, and 28 that have a nontrivial automorphism of order a power of 3. If n = 18, 20, or 22, then d = 8; if n = 26 or 28, then d = 10. By Lemma 1, we may assume that the diagonal elements of the automorphism groups of the codes we are considering are ( w l ) . Hence by Section 11, we may assume that if 8 is such a code, Aut(&) contains either a Type I or Type I1 automorphism. To have a Type I1 automorphism, 31n and hence a Type I1 automorphism is possible for the values of n under consideration only when n = 18.

Before proceeding with the classification, we give two results that will narrow the choices of possible Type I automorphisms for the values of n we are examining. The first is general.

Lemma 4: Suppose B is an [n,n/2,d] self-dual quaternary code with Type I automorphism M with c, fo , f l , f 2 as in Section 11. Then for 0 I k I 2,

b) if f k 2 d , c + f k 2 2 d - 2 . a) if f k I d - 1 , C 2 fk,

Proof As 8 is self-dual, any d - 1 coordinates of k =

E,(M)@E,(M)@E,(M) are independent. As E k ( M ) is 0 on the fixed points of E,(M) , E , ( M ) where {i , j , k ] = (0,1,2}, any d - 1 or fewer fixed point coordinates of E k ( M ) are independent. Hence dim E k ( M ) 2 min(d - 1, f k ) . As dim E k ( M ) = (c + fk)/2

0 If x E 4, wt(x) denotes the weight of x as a vector in F t . If

k has a Type I automorphism M and if x E E k ( M ) , let x * E E k ( M ) * be the vector associated with x as in Section 11; denote the weight of x * as a vector in Ft'fk by wt(x*). Notice that if x has t nonzero entries in fixed point coordinates, wt(x)= 3(wt(x*)- t ) + t . The self-dual quaternary codes of length n I 14 are classified in [ll]. These codes will turn out to be the possible codes E k ( M ) * and the notation of [ l l ] will be used.

Lemma 5: Let B be an [n , n /2, d ] extremal self-dual quaternary code with Type I automorphism M and n = 18, 20, 22, 26, or 28. Then for 0 I k I 2,

by Section 11, the result follows.

a) c 2 4,

c) if c + f k = 6, c 2 4 if n = 18, 20, or 22 and c 2 5 if n = 26 or 28, and Ek(M)* = E,,

d) if c + f k = 8, c 2 6 if n = 18, 20, or 22 and c 2 7 if n = 26 or 28, and Ek(M)* = E,, and

e) if c + f k = 10, Ek(M)* = E,, or B,,; if Ek(M)* =E,,, c 2 5 if n=18 , 20, or 22 and c 2 9 if n = 2 6 or 28; if Ek(M)* = B,,, c 2 6 if n = 18,20, or 22 and c 2 8 if n = 26 or 28.

Proof: If x E E k ( M ) with Wt(x*) = 2 ,2 I wt(x) I 6, a contradiction. Hence Ek(M)* must all have minimum distance 4 or

b) C + f k 2 6,

1


greater. By [11], this implies b), and if c + f k = 6 , 8, or 10, Ek(M)* are only the codes listed in c), d), and e). If x E E k ( M )

TABLE I ASSOCIATION BETWEEN F4, I,, I,, AND 1,

is such that wt(x*) = 4 and if x * has t nonzero entries in fixed Entry In 1, I7

0 0 0 0 0 0 0 0 0 0 point coordinates, then t I 2 if n = 18, 20, or 22 and t I 1 if n = 26 or 28 as wt(x>= 3(wt(x*)- t ) + t . By using this fact, it is 1 1 1 1 1 W O l o w

0 0 0 1 w J w 1 0 1 w

w w w w l J - _ _ _ w w w

a straightforward examination of the weight 4 vectors in E,, E,, E,,, and B,, to find the maximum number of coordinates that could be fixed points. Such an examination vields the results of c), d), and e). We only need to verify a). If c 5 3, by b), c), d), and e), c + f k 2 12 implying by Lemma 4, c + fk L 2d - 2 2 14

0 for all k . So fo + f l + f 2 2 33, a clear contradiction.

Theorem 4: Let k be an [ n , n / 2 , d ] extrema1 self-dual quaternary code with Type I automorphism M . Then we may assume one of the following forms occurs.

a) If n = 18, either 1) c = 4, fo = f l = f2 = 2 and E k ( M ) * 2 E, for 0 I k I 2,

2 ) c = 5, f o = f, = f2 = 1 and Ek(M)* = E, for 0 I k I 2,

3) c = 6, f o = f, = f2 = 0 and E k ( M ) * E, for 0 I k I 2. b) If n = 20, c = 6 , f o = 2, f l = f 2 = 0, Eo(M)* = E , and

Ek(M)* = E, for 1 I k I 2. c) If n = 22, either

or

or

1) c = 5, fo = 5 , f l = f 2 = 1, Eo(M)* 2 E,,, and E,q(M)*

2 ) ~ = 6 , f o = 4 , f 1 = f 2 = 0 , Eo(M)* E10 or Bio, and = E, for 1 I k I 2, or

Ek(M)* = E, for 1 I k 5 2 , Or

E , for 1 I k I 2. 3) c = 6, fo = 0, f l = f 2 = 2, E,(M)* = E,, and E,(M)* 2

d) If n = 26, c = 8, fo = 2, f l = f 2 =o , Eo(M)*= B,o, and

e) If n = 28, c = 8, fo = 0, f l = f 2 = 2 , Eo(M)*= E,, and Ek(M)* = E, for 1 I k I 2.

Ek(M)* = B1o.

Proof Suppose first that c = 4. Then either c + f k 2 12 for all k , or f k = 2 for some k and n = 18, 20, or 22 by Lemma 5. If c + f k 2 12 for all k , by Lemma 4, c + f k 2 14 for all k and fo + f l + f z 2 30, which is impossible. If n = 18, 20, or 22 since f o + f l + f2 I 10 and fo, fl , f2 2 2, c + fk I 10 implying by Lemma 5 , f o = f l = f2 = 2; hence only al) can occur with c = 4.

Suppose that c = 5 . If n = 26 or 28, for some k , fk > 1. But if n = 26 or 28, by Lemma 5 f k # 3,5 and by Lemma 4 if f k > 5, c + f k 2 18 which is impossible as f o , f l , f 2 are all odd and f o + f l + f 2 = n - 3 c ~ 1 3 . If n=18 , 20, or 22 by Lemma 5, f k # 3. So the only possibilities are (fo,fl , f,} = {1,1, 1) if n = 18 and (fo,fl,f2}=(1,1,5} if n = 22. By the discussion of Section 111, we may assume any order for f0 , f l , f2 and hence a2) and cl) hold by Lemma 5.

Suppose that c = 6. As fo , f l , f 2 are even, by the discussion of Section 111 and by Lemma 5, we have only the possibilities a3), b), c2), and c3) when n = 18, 20, or 22. If n = 26 or 28, for some k , f k 2 2. But if n = 26 or 28, by Lemma 5 fk # 2,4 and by Lemma 4, if f k > 6, c + f k 2 18 which is impossible as f o + f l + f 2 = n -3c I 10. So fo , f l , f, are either 0 or 6 when n = 26 or 28, but this contradicts fo + f l + f 2 = n - 3c.

Suppose that c = 7. As f 0 , f l , f 2 are odd, n # 18, 20, or 22. If n = 26 or 28, f k # 3 by Lemma 5. The only possibility is n = 28

[11] and must be E,eE,, E,,, C12, D,, or F12. But examining the vectors in these codes of weight 4 shows that at most 2, 1, 2, 2, or 3 coordinates respectively could be fixed points.

and (fo,fl , f2) = (1,1,5}. If f k = 5 , Ek(M)* is a [12, 6, 41 code Of

Suppose that c = 8 . Then n = 26 or 28 and certainly by the discussion of Section 111, we have either d) or e) by Lemma 5, or n = 28 and {f,,f,,f,} = (0,0,4}. But this latter case is eliminated as the case (f,, f l , f,} = (1,1,5} was eliminated.

Finally, as 3c I n , only c = 9 remains and hence n = 28; but 0

We now examine the individual values of n. When examining codes with a Type I automorphism, we will assume the matrices have columns ordered with cycle coordinates on the left, followed by the fa columns of F,u, the f l columns of Fw, and finally the f 2 columns of F,2. We will give generator matrices for Eo(M), E , (M) , and E,(M). The cycle coordinates will be written to the left of I in each matrix and will be abbreviated using the appropriate field Io , I , , or I , . For example, the entry w in E,(M) to the left of I will represent w i , ( X ) = w + WX + X 2 = wWl. For easy reference the complete association is given in Table I .

then fo , f l , f 2 are odd, which is impossible.

A. n=18

Theorem 5: Let 8 be a self-dual [18, 9, 81 quaternary code with a nontrivial automorphism of order a power of 3. Then 8 has a Type I automorphism M with c = 5, f o = f l = f2 = 1; k= E,(M)@E,(M)@E,(M) where

1 0 0 1 1 1 1 0 0

0 0 1 1 W I w O O

1 0 0 1 1 I O 1 0

o o l l w l o w o 0 1 0 1 w I 0 0

1 0 0 1 w I O O 1

0 0 1 w 1 I 0 0 1

0 1 0 1 w I W 0 01,

0 1 0 w w I 0 0 W].

and

The code B is S,, of [l].

Proof: First assume k has a Type I1 automorphism M . Then k* is a [6,3] self-dual code under ( 2 ) over FM. A primitive element of F64 is a = W + w X 2 = W O O . If x E 6 is associated with x * E 8*, wt(x) denotes the weight of x in Fd8, and W ( x * ) denotes the weight of x* in FA. If there exists a vector x * E B* with 0 < W ( x * ) I 3, by scaling we have a vector y* = cx* with some entry a" = 100 and 0 < W(cx*) I 3. But then 0 < wt(y) I 7, a contradiction. So B* is a [6,3,4] MDS code. By Theorem 2, we may permute the columns of 6*, scale columns by independent powers of a63/9 = a' (this operation corresponds to applying W * of Theorem 31, and replacing a by a,' (this corresponds to applying (S)). We find the possible forms of vectors x* E B* where W ( x * ) = 4 with some coordinate a'. Using the inner product and avoiding low weight vectors, it is easy to verify that x * = a0a1a2a500 or a0a3a5a600 up to coordinate permutation, scaling columns by powers of a', or replac-


ing a by a2'. Note that a5(a0,a1,a2,a5,0,0)=(a5,a6,a7~a0, a7.a3,0,0). Using these facts and the inner product, we obtain the following:

(3) o o ao a4* a' a'

or

(4) o o a0 a3* ak am

By taking the inner product of the first and third rows, we get a J = a24 + a55+1 and Also in order for the third row to be equivalent to an x * , we must have (i, j ) = (1 + 7u,4+7u) or (3+7u,5+yu}, and (k,m)={2+7u,3+7u} or (1+ 7u, 6 +7u} with 0 I U, U I 8. By hand and by computer, the only solutions are (i, j ) = (12,38), (43,32), (57,251, (59,331, or (61, lo), and ( k , m ) = (1,6), (6,15), (22,55), (44,311, (51,3), or (62,l). However in each case a low weight vector in d is found, by computer and verified by hand, from a combination of the first and third rows of (3) and (4). So d cannot have a Type I1 automorphism.

Suppose d has a Type I automorphism M . By Theorem 4 we have a). First assume c = 4, f o = f l = f 2 = 2. By row reducing, scaling columns, and possibly interchanging fixed points of Fw", we may assume

1 0 0 1 l 1 1 0 0 0 0 gen(E,(M))= 0 1 0 1 I w U 0 0 0 0 I 0 0 1 1 I U w 0 0 0 0

By row reducing, applying elements of W of Theorem 1 (which act trivially on Eo(M)), scaling the fixed points of FWl, and possibly interchanging the fixed points, we may assume

1 0 0 1 I 0 0 1 1 0 0

0 0 1 1 I 0 0 W w 0 0 By row reducing and scaling the fixed points of F w 2 , and possibly interchanging these fixed points, we may assume

gen ( E2( W )

0 0 1 d 1 0 0 0 0 bdW bdw

1 0 0 b J 0 0 0 0 1 1 0 c I O 0 0 0 bcw

where b, c , d E (1, w , W}. (Note that the actions done to obtain gen(El(M)) do not affect gen(E,(M)), and those done to obtain gen(E,(M)) do not affect gen(E,(M)) or gen(E,(M)). This is the strategy that will be followed throughout this section.) Applying elements from Z: ZfuL3( ST) Zf IZ fZ4W we can by hand reduce the number of cases to check; all yielded nonzero vectors in 6' of weight less than 8. (This was verified by computer.)

Now assume c = 5, f o = f l = f z = 1. As before, we may assume

1 0 0 1 1 l 1 0 0 o 1 o 1 a I (Y o o 0 0 1 1 c r l a 0 0

1 0 0 1 1 1 0 1 0 0 1 0 1 p 1 0 o , 0 0 1 1 p I 0 p O 1

and

where a ,p ,yE(w,U) and b,c,d,eE(l ,o,U}. Two of a , p , y must be equal. By applying elements of 9= ( S , , ~ S , S ~ , ~ , ~ K - ~ ) , rescaling columns, and applying elements of W, we may assume a = p .

By applying ST if necessary, we may assume a = p = w . Ap- plying elements from 2: X , 9 ( S T ) XfIZ,4W we can reduce the number of cases to check. The only case, up to equivalence under these elements, yielding an extremal code is the one given in the statement of Theorem 5.

Finally assume c = 6, f o = f l = f 2 = O . As before, we may assume

and

0 0 1 f bcfr bdfy

0 where b,c, d,e, f E (1, w , W} and y E { w , W). Reducing the number of cases as before, no extremal codes were found.

Corollary 2: If d is a self-dual [18, 9, 81 quaternary code with a nontrivial odd order monomial automorphism, d is S,, of [l].

0

I d I l O O b c gen(E,(M)) = 0 1 0 e k e y bdei , I

Proof By Theorem 5 of [9] and Theorem 5.

B. n = 2 0

Theorem 6: Consider all self-dual [20, 10, 81 quaternary codes with a nontrivial automorphism of order a power of 3. All such codes have a Type I automorphism M with c = 6 , f o = 2, f l = f 2 = 0. All such codes are equivalent to one of two inequivalent codes 6= E,(M)@E,(M)@E,(M) both with

1 1 0 1 0 1 0 I 1 0 1

1 0 0 1 1 1 I 0 0

0 0 1 1 W w I O O

and

0 0 1 w oa W a

1 0 0 1 0 0

gen(E2(M))= 0 1 o w wcu uz I one code has a = U and the other has a = U.


1 1 1 0 0 0 l 1 0 0 0 0 0 1 1 0 0 I 1 1 0 0

0 0 0 0 1 1 I 0 0 1 1 - w w 1 1 1 1 I o o o o

= o 0 0 1 1 0 1 0 1 1

Proof: Only Type I automorphisms satisfying b) of Theorem 4 need be considered. In that case, since Aut@) is triply transitive, by scaling the columns, we may assume gen(E,(M)) is as stated in the theorem. By row reducing, applying elements of F, and applying ST, we may assume gen(E,(M)) is as stated. By row reduction

l O O b c d I 0 0

0 0 1 f EcfG Edfa I 0 0 0 1 0 e k e a EdeZ I 0 01.

Applying all elements of 8r8 fod(s7 )8 f ,8 fz (~1 ,2~ )~ by computer and checking minimum weights of representatives of each class under this group, we found only the two extremal codes stated in the theorem. Using Theorem 2 we found that in the case where 8 has b = 1, c = d = e = f = CY = o the Sylow 3-subgroup has order at least 27 but in the case where 8 has b = 1, c = d = e = f = o , a = O the Sylow 3-subgroup is ( M , o I ) of order 9. So the two codes are indeed inequivalent. (Theorem 1 also guarantees this once we know the Sylow 3-subgroup is (M, wl) for one of the two codes.) 0

0 , ( 5 )

1 0 0 0 0 I 0 w w w w 0 0 o 1 o o o I w w o o w o o

= 0 0 1 0 0 l w w w 0 0 0 0 , o o o 1 o ( o w w o w o o 0 0 0 0 1 I w w w w w 0 0 -

- - - - -

TABLE I1 PARAMETERS FOR gen(E,(M,)) WHEN n = 22

= O

Code Order Number b c d e a Sylow3

- - 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 I w 1 0 0

0 0 0 0 1 0 l 0 0 0 1 - 0 1 w 0 1 0 I w 0 w 0

0 0 1 1 1 1 0 0 1 0 , ( 6 )

1 1 1 1 1 0 9 2 l l l w w 9

l w o w w 9 9

3 4 1www0

l l l l W 27 81

5 6 l l w w W

TABLE 111 PARAMETERS FOR gen(E,(M,)) WHEN n = 22

Code Number

7 8 9

10 11 12 13 14 15 16 17 18

Order gen(Eo(M2) ) b c d e f a Sylow3

(5) l w o w o o 9 (5) l 0 w w l o 9 (5) l l 0 0 w W 9 (6) l l W l w o 9 (6) l l w o l 0 9 (6) l l w 0 l O 9 (5) 1 1 1 o 1 0 2 2 7 (5) 1 1 1 o w 0 2 2 7 (5) W W W o 1 0 2 2 7 (6) l l w w l w 27 (6) w l l 0 l O 27 (6) w l 0 0 l W 81

and gen ( E2(Ml))

l O O b c 1 0 0 0 0 0 0 1

1 l O O b c d ( 0 0 0 0 0 1 0 e bcea bdeG I 0 0 0 0 , 0 0 1 f bcfZ bdfa I 0 0 0 0

where the sets of parameters in gen(E2(Ml)) are found in Table 11. The column “order Sylow 3” is the order of the Sylow 3-subgroup of the automorphism group of the code.

where the form of gen(E,(M,)) and the sets of parameters in gen(E,(M,)) are found in Table 111.

Theorem 8: Consider all self-dual [22, 11,8] quaternary codes Theorem 9: Consider all self-dual [22, 11,81 quaternary codes with a Type I automorphism M2 with c = 6, fo = 4, with a Type I automorphism M3 with c = 6, f, = 0, f l = f 2 = 2. f, = f 2 = 0. All such codes are equivalent to one of 12 inequiva- All such codes are equivalent to one of 24 inequivalent codes

-~

- I-


TABLE IV PARAMETERS FOR gen(E,(M,)) WHEN n = 22

Code Order Number gen(E,(M,)) b c d e f Sylow3

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

(7) l l w w l 9 (7) l l w w w 9 (7) l l w w 0 9 (8) 1 1 1 1 0 9 (8) 1 1 1 0 0 9 (8) 1 1 0 1 0 9 (8) l l w w l 9 (8) 1 1 0 0 0 9 (8) 1 1 0 1 0 9 (8) l l 0 O w 9 (8) l 0 w l l 9 (9) 1 1 1 1 0 9 (9) 1 1 1 1 0 9 (9) 1 1 1 0 0 9 (9) 1 1 0 0 1 9 (9) 1 1 0 1 0 9 (9) l w w l w 9 (9) l w w 0 w 9 (9) l w 0 l w 9 (9) l w J l 0 9 (9) 1 0 0 1 1 9 (9) 1 0 0 0 0 9 (7) 1 1 0 0 0 27 (7) 1 1 w 0 1 81

8= Eo(M3)@El(M3)$E,(M3) with

1 1 0 0 1 1 1 1 0 0 0 0 o 1 o 1 w w I o o o o , 0 0 1 1 ' 0 w I O 0 0 0

and

l O O b O c I O 0 1 O l O d O b c d I O O O 1 0 0 1 e O O I 0 0 b e Ze 0 0 0 0 1 f I O O Z f b & T f

or

gen ( M3))

l O O O b c I O 0 1 O l O O d & c d l O O O 1 O O I O ~ o 1 0 0 b e iie O (8)

O O l O e b c e I O O 0 de 1 (9)

=I 0 0 0 1 0 f l o o q b ~ ~ f

=I 0 0 0 1 0 f l o o q b & T f

or

gen ( M3))

l O O O b c 1 0 0 1 O l O O d 0 I O O b d 1

where the form and set of parameters in gen(E,(M,)) are found in Table IV.

Theorem 10: The only equivalences between the codes 1-42 of Tables 11-IV are as follows:

a) 16 and 41 are equivalent, b) 5 and 17 are equivalent, and c) 6, 18, and 42 are equivalent.

There are 38 inequivalent [22, 11, 81 quaternary self-dual codes with a nontrivial automorphism of order a power of 3.

We proceed with the proofs of Theorems 7-10.

Proof: The generating matrices for E,(M,) in Theorems 7, 8,and 9 are obtained from Theorem 4cl), 21, and 3) respectively. To obtain gen(E,(M,)), we first note that if U* E E,(M,)* with wt(u*) = 4, at most 2 nonzero coordinates of U* may be fixed points of s",~; otherwise 0 < wt (U) I 6. We illustrate the process by showing that gen(E,(M,)) is ( 5 ) when E, (M, )*=E, , ; the other cases are similar. E,, contains 10 weight 4 vectors that are obtained when their supports are any pair of the unordered pairs {a, b},{c, d},{e,f},{g, h},{ i , j } where a, b; . ., j are the 10 coordinates. We would obtain a vector U* E E,(M,)* with wt ( U * ) = 4 and 3 or 4 nonzero coordinates in S,U if any of the previous 5 pairs is contained in F,u. So by permuting the cycle coordinates and Y,O, we may assume the 5 pairs are {1,2}, {3,7},{4, 8), (5,9},{6, lo), where 1,2, * * , 6 represent cycle coordinates and 7,8,9,10 fixed points. By scaling columns and using self-orthogonality we obtain the first four rows of (5). In E,, there are vectors of weight 6 that are nonzero on any chosen pair and zero on exactly one of the two coordinates in the remaining 4 pairs. We choose one of these vectors U * that is nonzero on {1,2). By orthogonality with rows 1-4 of (9, adding combinations of rows 1-4 to U , scaling U , and possibly interchanging columns 1 and 2, we obtain row 5 of (5).

To obtain gen(E,(M,)) we may apply elements of Aut(E,(M,)*), scale and permute fixed points of F-1, and apply elements of 7, and apply s. (See Section 111.) These actions fix

E,(M,). For example, in the case of E,(M,), row reducing E,(M,)* = E, and applying elements of W (which act as scalars on the cycle coordinates), we obtain either gen(E,(M,)) in Theorem 8 or

gen ( El( M2) 1

1 [ 0 0 1 1 0 W I 0 0 0 0

1 0 0 1 1 1 I 0 0 0 0 = o 1 0 1 w w I O 0 0 0 . ( 1 0 )

If gen(E,(M,)) is (5), as f = (5,6),(9,10) E Aut(E,(M,)*), applying T to (!O) gives the form in Theorem 8. If gen(E,(M,)) is (6), as T = (7,8)(9,10) diag ( U , W, 1, w , W, 1 , 1 , 1 , 1 , 1 ) ~ E Aut(E,(M,)*), we apply T = Tu,%,u~u5s to (10) (which simply acts as conjugation) to obtain the form in Theorem 8.

The forms of gen(E,(M,)) for i = 1 and 2 are immediate. By applying elements of 8TZf, ,9( s r ) Z f l Z f i that fix gen(E,(M3)) and gen(E,(M,)), we obtain the forms (71, (81, and (9) for gen (E2(M3)).

Now suppose 8 and 8' are codes with the same automorphism M of Type I. For notational purposes, we will say 8' = d (8' is related to 8) if there exists an N E

~ ~ ~ f o 9 ( ~ r ) 8 . f 1 8 f 2 ( s l r 2 ~ ) W s u c h that 6'= 8 N . We considered each case according to different forms of generator matrices. In the case of M = M , or M 2 , forms with a = w and a = U in gen(E,(M,)) are considered different, and if M = M2, forms with E0(M2)* = E, , or B,, are different. So there are 2 different forms when M = M , and 4 different forms when M = M,;

- 1


TABLE V G WHEN n = 22

Case 6

when M = M3, there are 3 different forms depending on whether gen(E,(M,)) is (7), (81, or (9). By computer we found representatives of each class under - for each different form. (We will see later that if 8'- 8 where 8 is extremal, then 8' and 8 have to be of the same form.) We found the representatives as follows. With each form we ordered the codes by ordering the parameters b, c, d , . . . lexicographically. To a coden 8, *using Corollary 1, we applied by computer elements of G ={TIT E

Z : Z f O 9 * ( s r ) } n Aut (E,(M)*) followed by elements of Zf,Zf2(s1,,s) followed by elements of 9Y. If we ever got a previous case with these maps, we considered the case done; otherwise we tested for low weight vectors. The application of these maps, using Corollary 1, is relatively fast. Since within a given form, gen(E,(M,)) is unique, elements of ZzxfogA*(sT) are determined uniquely by elements of the appropriate G. (See Table V.) Because E,(M,) is unique for each i, E,(M,)* = E, or E,, and the only diagonal elements of Aut(E,) and Aut(E,) are scalars, the map from Z : Z f o 9 * ( s r ) followed by the map from ZflZf2(slrZs) determines at most one "diagonal element" up to multiplication by wZ of 9'Y will preserve the form.

Before proceeding further, we describe how a low weight vector was found. No such vector could be zero on each cycle, because the only vector in E,(M,) that is zero on all cycles is the zero vector and E,(M,)+ E,(M,)+ E,(M,) is a direct sum. So if U E 8 has wt(u) I 6, U * has some nonzero entry 100, 110, 100, GO, 111, 110, l l W , l w 0 , or 1Ww on some cycle by scaling and applying M, if necessary. We stored representatives of all one- dimensional subspaces of E,(M,)*, E,(M,)*, and E,(M,)*. For each cycle, we combined all possible vectors of wt I 6, one from each E,(M,)*, j = 0,1,2, so that the vector was 100 on the cycle. If no low weight vector was found, for each cycle, we combined all possible vectors of wt I 3 on cycle positions (higher total weight possible), one from each E,(M,)*, j = 0,1,2, so that the vector was 110, lw0, or lU0 on the cycle position. If no low weight vector was found, for each cycle, we combined all possible vectors of wt I 2 on cycle positions (higher total weight possible), one from each E,(M,)*, j = 0,1,2, so that the vector was 111, 110, llW, low, or 1Ww on the cycle position. This would yield a low weight vector if one existed. This process proved to be relatively fast.

We found representatives of each class of each form under - . When an extremal representative was found, the subgroup of its automorphism group contained in

was computed. Using Theorem 2, we can decide whether or not (M,,wZ) is a Sylow 3-subgroup of the automorphism group. In what follows, let < be code #i in Tables 11-IV. Also let 4= NAUtc8,,,(( MJ,wZ)) where MI is the appropriate automorphism of 4. By Theorem 2, Jt: ~ Z z Z f " ~ ( s T ) ~ ~ l Z f 2 ( s l , 2 ~ ) ~ .

A matrix M E A?,, of order 3 will be of type MI if for some TEA?;, T-'MT E (M,,wZ). So if M, Mi E Aut(&) with M of type M,, then using the second paragraph of Section 111, 8 is equivalent to a code 8' with M, E Aut (6"). Note also by examining eigenvalues and cycle structures, there is no T E k g with T-'( Mi, wZ)T = ( M , , w Z ) when i # j .

The results for MI were as follows. For a = w , there were three unrelated extremal codes, ~??,,8,,8~ of Table 11. For (Y = W, there were four unrelated extremal codes 84, 8s, g6 plus an additional code. Using Table 11, we found Aut(<) for 1 I i I 4 have Sylow 3-subgroups (M,, w Z ) , and Aut (-es) and Aut(&,) have Sylow 3-subgroups of order at least 27. We also verified that if 4 and 4. are in this list but one had a = w and the other a = W, 6 # q. Hence by Theorem 1, < for 1 I i I 4 are inequivalent. gS and 86 are not equivalent to any 4. with 1 I i I 4 by order of the Sylow 3-subgroup of and Aut(86). Later we will see why the Sylow 3-subgroups of Aut(gS;) and have the order given in Table 11. The other code 8 not listed had acother element M E Aut(&), M E ( M , , w Z ) with M of type M,. By changing basis to get M in the form of M,, 8 was found to be equivalent to &S.

The results for M, were as follows. For gen(E,(M,)) of form (5) and (Y = w , there were two unrelated extremal codes, TZ?7 and 8,. For gen(E,(M,)) of form (5) and (Y = W, there were eight unrelated extremal codes, 89, 8,3, 814, 81s and four other codes. Aut(<.) has Sylow 3-subgroup ( M2,wZ) for 7 I i I 9, and the automorphism groups of the remaining codes have Sylow 3-subgroups of order at least 27. We verified that if 6. and 5. are in the list but one had (Y = w and the other a = W, < # 3. Hence by Theorem 1, 87, 8,, are inequivalent. 813,&14,81s and the other four are not equivalent to any 8,, 8x, 89 by order of the Sylow 3-subgroup of their automorphism groups. Using Theorem 2, we found that the Sylow 3-subgroups of N, for 13 I i I 15 and of the associated normalizer of the unlisted four codes have elements of order 3- and the nonscalar elements among these all are of Type M,. Using these elements, by changing basis, we found three of the unlisted codes are equivalent to gI3 and the other to gI4. We verified that ~,3,814r81s are inequivalent as follows. We found the 330 supports of the weight 8 vectors in each of these three codes. In 813, there are several pairs of coordinates contained in exactly 49 of these supports; there are no pairs of coordinates contained in exactly 49 of the supports in 814 or 81s. The supports in 81s form a 2-(22,8,40) design, but do not in 814. So 813,814,815 and hence all <., i E (7,8,9,13,14,15} are inequivalent.

For gen(E,(M2)) of form (6) and a = w , there were two unrelated extremal codes, 8,, and g,,. For gen(E,(M,)) of form (6) and a = W, there were four unrelated extremal codes, 811,~l~,817,81x. We verified that if 4 and 4. are in this list but one had a = w and the other had a=W, <.#%. By Corollary 1 as E, , f. B,,, codes with gen(E,(M,)) of form (5) are unrelated to codes with gen(E,(M,)) of form (6). 8,,, 8,,, gI2 have Sylow 3-subgroup (M,, w Z ) in their automorphism group and hence are inequivalent to each other and to any <, iE{7,8,9,13,14,15} by Theorem 1 and orders of Sylow 3-subgroups. Let < be the Sylow 3-subgroup of &', for 16 I i I 18. In each case < - (M,, 0' ) contains only elements of order 3 and none are of Type M,. Suppose 9; is a Sylow 3-subgroup of Aut(<) with 4 ~ 9 ~ . If this containment is proper, by [6], there is an element N €9, r& normal- izing <. But as N-'M2N must be of Type M2 and in 8, N-'M,NE(M,,wZ) as ;4-(M2,wZ) contains no such elements. Hence N E Jt: by Theorem 2, a contradiction. So < is a full Sylow 3-subgroup of Aut(<). SI, and Y17 have order

- I


27; S,, !as order 81. S,, - ( M 2 , w l ) contains only elements o-f Type M3; S17 - ( M 2 , o l ) contains only elements- of Type M,; S,, - ( M 2 , o l ) contains only elements of Type M , or M3. Thus, as 916,S17,S18 cannot be conjugate by an element of A?$, codes 8167 TZ,~, 8118 are inequivalent. These also cannot be conjugate to the Sylow 3-subgroup of Aut(<), 131i115 as these Aut(<) contain a 3-subgroup o,f order 27 containing ( M , , w l ) and only elements of Type M,. Hence all codes in Table 111 are inequivalent. By using elements of S17 - ( M 2 , o l ) and S18 - ( M,, col) and changing basis, we found B17 is equivalent to 8’ and 81s to 8,. Hence d5 and 8, are inequivalent and their automorphism groups have Sylow 3-subgroup orders 27 and 81 respectively. As ( M l , w l ) and ( M 2 , w l ) are not conjugate in A?&, codes <, 1 I I I 4, are inequivalent to any in Table 111. Thus the only equivalences between Tables I1 and 111 are 85 to 7f17 and 86 to d1,.

The results for M3 were as follows. For gen(E,(M,)) of form (7), there were five unrelated extremal codes 819,820, 821, B41, 842. For gen(E,(M,)) of form (8), there were eight unrelated extremal codes < for 22 I i I 29. For gen(E,(M,)) of form (9), there were eleven unrelated extremal codes < for 30 I i I 40 plus one other code. If < and 4 are in this list but with different forms, say (7) and (8) respectively, and < N = 4 with N E 8 ~ C . , u ~ ( s ~ ) 8 , 1 8 , z ( s 1 , 2 ~ ) ~ , then because the form of (7) or (8) is determined solely by the supports of U * E E2(M3)* where wt(u*) = 4, N must send all codes of form (7) to codes of form (8). This implies that each of form (7) and (8) would yield the same number of unrelated extremal codes, which is not the case. This also holds for forms (7) and (91, and forms (8) and (9). Thus if < and $ are in the list but have different forms, <+ 4. Aut(4:) has Sylow 3-subgroup ( M 3 , 0 1 ) for 19 I i I 40 and so by Theorem 1 the 6 are all inequivalent. The unlisted code, 841, and 84, have an element in the normalizer of (M, , u l ) in their automorphism groups of Type A&. By changing basis, the unlisted codes and 8 4 1 are found to be equivalent to B16, and 842 is equivalent to Bl,. By sizes of the Sylow 3-subgroups of Aut(<), all < in Table IV are now inequivalent. As ( M 3 , 0 1 ) is not conjugate to (Ml, u l ) or ( M 2 , 0 1 ) , < for 19 I i I 40 is not equivalent to any code in Table I1 or 111. This completes the proof of Theorems 7-10. 0

have elements of order 5. Hence the codes 83 and 85 (and the equivalent code 817) have an automorphism of order 5, and so are among codes of Theo- rem 7c) of [9].

We remark that M3 and M5

D. n = 2 6

Theorem 11: There does not exist a self-dual [26,13,10] quaternary code with a nonscalar odd order monomial automorphism.

Proofi By Theorem 7 of [9], if a self-dual [26,13,10] quaternary code 8 exists with a nonscalar monomial automorphism of odd order, the automorphism is of order of power of 3. By Lemma 1, 8 must have a Type 1 automorphism M . By Theorem 4d) and carrying out arguments similar to previous values of n, we have c = 8, fa = 2, f l = f2 = 0 and

! I o 1 w o o 1 w w I o o

1 1 1 1 0 0 0 0 I 0 0 0 1 0 W 0 0 0 0 I 1 0

0 0 0 0 0 1 w 0 I 0 1 gen(E,(M))= 0 0 0 0 1 1 1 1 I 0 0 .

We notice that, if x* E E,(M)* and y* E E,(M)* where i # j with wt(x*)=wt(y*)=4, and x* and y* have the same supports, then wt (x + y) = 8, a contradiction. Also notice that

6 = {FIT€ H:Plir9*(sr)}nAut(E,(M)*)

=( (1,3)(2,4),(1,2,3)diag( l , l , l , l ,U ,U ,W, W , o , U ) ,

(1>5)(2,6)(3,7)(4, 8)(9,10), (1,2)(5,6)7) . Applying elements of 6 and W, and using the observation on Wt4 vectors, we get

gen ( El( M I )

1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0

or gen(El(M)) is (11) with the last row replaced by

1 0 1 1 0 0 0 1 I O 0.

Applying elements of 6 which fix each of the forms of gen(E,(M)*), we find that there is only one set of supports for U * E E2(M)* where wt(u*)= 4 for each form of gen(E,(M)). An element of C y 8 , 9 * ( S T ) C . ~ ~ Z , ~ S ~ , ~ S fixing E,(M) swaps these two pairs of ( E l ( M ) * , E 2 ( M ) * ) by sending E,(M)* from the first pair to E,(M)* of the second pair, with {i, j}=(l,2). So by possibly reapplying elements of Y, we may assume gen(E,(M)) is (11) and

gen ( E2( W ) 1 0 0 0 0 b c d I 0 0

0 0 0 1 h 0 zcefh bdefh I 0 0

where b, c, d , e , f, g , h E {l, o, W). As in the case n = 22, we applied elements from 8 ~ C , u ~ * ( s ~ ) C , l ~ f z ( s l , 2 s ) ~ ~ to reduce the number of cases to check. No case yielded an extremal code.

0

E. n=28

Theorem 12: Let B be a self-dual [28,14,10] quaternary code. with a nontrivial automorphism of order a power of 3. Then 8 has a Type I automorphism M with c = 8, fa = 0, f l = f2 = 2; 8 = E,(M)@E,(M)@E,(M) where

gen ( Eo( M ) )

0 0 0 0 1 1 1 1 1 0 0 0 0 ’

1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 1 =I 1 0 1 0 1 0 1 0 1 0 0 0 0

gen ( El( M I )

! I 0 0 0 0 0 1 0 w I 0 1 0 0

1 0 0 0 1 0 1 I l l 1 0 0 0 1 0 0 1 0 w w I 0 w 0 0

= 0 0 1 0 1 O o o I ~ o O O , 0 0 0 1 0 0 0 0 l 0 1 0 0

1216

and

l 0 0 0 0 l l w l 0 0 l l 0 l 0 0 0 W l w l 0 0 w l

= 0 0 1 0 0 0 1 W l 0 0 0 W . 0 0 0 l 0 w l w l 0 0 G l 1 0 0 0 0 1 0 1 1 I 0 0 0 w _

IEEE TRANSACHONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

- - l 0 0 O O b c d 1 0 0 1 1 O 1 O O O e Zcep &e I 0 0 &ep be 0 0 1 0 0 f 0 b d f r I 0 0 0 6 f -y . 0 0 0 1 0 g z c g p 6dg I 0 0 bgp bg

- O O O O l h 0 b d h y I 0 0 0 bh?

- -

- - -

We divide both (12) and (13) into four cases depending on the values of P,y. By applying elements of

s: Zfllg* ( S T ) XflZf2( , 2 S )g’y,

the 16 cases, made up of gen(E,(M)) and gen(E,(M))’with the eight cases for gen(E,(M)) from (12) and (13), actually reduce to eight cases: gen(E,(M)) with p = y = w in (121, /3 = 7 = w in (12), /3 = y = w in (131, p = y = W in (131, and gen(E,(M))’ with p = y = w in (121, p = y = W in (121, p = y = o in (131, p = 7 = w in (13). Each case has 3’ codes to check. As in previous values of n , applying elements from 2: Zfo9*(sT) Zf,C f2( sl,*s > 9 W in each of these eight cases, we reduce the number of codes to check. The only external code, up to equivalence, occurred for gen(E,(M))with b = c = f= h = 1, d = g = w , e = W in (12). 0

TABLE VI NUMERICAL RESULTS

n

18 18 18 20 22 22 22 26 28

c f ” f l f 2

4 2 2 2 5 1 1 1 6 0 0 0 6 2 0 0 5 5 1 1 6 4 0 0 6 0 2 2 8 2 0 0 8 0 2 2

= Classes Extrema1 = Classes

4 8

12 31 11 68 85

306 2994

0 1 0 2 7

16 25 0 1

V. SUMMARY

We conclude with some information about the number of codes where we actually had to locate a low weight vector. The number of codes we checked was the number of classes under = as in the n = 22 case; that is, &= 8 if there exists an N E c ~ Z f u g ( S T ) Z f l Z f 2 ( S , , , s ) W such that e= 8N. (In the case n = 18, the ( S , , ~ S ) could be replaced by (sl ,2s,s, , l ,2K-1), but this proved to be unnecessary in proving Theorem 5. ) The results are summarized in Table VI. Except in the case n = 22, the extremal classes were actually inequivalent.

There are some open questions about these lengths. In [9] for lengths n = 20, 22, and 28, extremal codes with automorphisms of prime order p > 3 were found; what is the equivalence between those codes and the ones of Theorems 6, 7, 8, 9, and 12? For lengths n = 18, 20, 22, and 28, are there extremal codes which do not have a nontrivial automorphism of odd order? In particular, is S,, the only self-dual [18,9,8] code? Finally, is there a self-dual [26,13,10] code? (See [51J

REFERENCES

[I] Y. Cheng and N. J. A. Sloane, “The automorphism group of an [18,9,8] quaternary code,” Discrete Math., vol. 83, pp. 205-212, 1990.

[2] 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.

[3] -, “Monomials of orders 7 and 11 cannot be in the group of a (24,12, IO) self-dual quaternary code,” IEEE Trans. Inform. The- ory, vol. IT-29, no. 1, pp. 137-140, Jan. 1983.

[4] 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, no. 2, pp. 312-322, Mar. 1979.

[5] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York: Springer-Verlag, 1988.

[61 D. Gorenstein, Finite Groups, 2nd ed. New York: Chelsea, 1980. [71 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, no. 3, pp. 486-493, May 1988.

[8] -, “On 3-elements in monomial automorphism groups of quaternary codes,” IEEE Trans. Inform. Theory, vol. 36, no. 3, pp. 660-664, May 1990.

[91 -, “On extremal self-dual quaternary codes of lengths 18 to 28, I,” IEEE Trans. Inform. Theory, vol. 36, no. 3, pp. 651-660, May 1990.

[lo] C. W. H. Lam and V. Pless, “There is no (24,12,10) self-dual quaternary code,” IEEE Trans. Inform. Theory, vol. 36, no. 5 , pp. 1153-1156, Sept. 1990.

[111 F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, “Self-dual codes over GF(41,” J . Combin. Theory A , vol. 25, pp. 288-318, 1978.

[I21 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes. Amsterdam: North-Holland, 1977.

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

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

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