IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992 I395

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0018-9448/92$03.00 0 1992 IEEE

[8] L. B. Wolfe and C.-I. Chang, "Source matching problems revisited," in Proc. Int. Cot$ Signal Processing '90, Beijing, China, Oct. 22-26, 1990, pp. 119- 122.

On Extrema1 Self-Dual Ternary Codes of Lengths 28 to 40

W. Cary Huffman

Abstract-The extrema1 self-dual ternary codes of lengths 28,32, and 36 with monomial automorphisms of prime order r 2 5, and of length 40 with monomial automorphisms of prime order r > 5 are enumerated. For each length and prime considered, we find all inequivalent extremal codes with an automorphism of that order.

Index Terms-Ternary codes, self-dual codes, extremal codes.

I. INTRODUCTION

In [4] and [5], we developed a general decomposition theory for self-dual linear codes V over a finite field F4 when V has a permutation automorphism of prime order r relatively prime to q. In Section 11, we summarize these results. Also in [4], [ 5 ] , [7], and [ 161 methods were developed for deciding, under certain conditions, whether two codes with the same automorphism are equivalent. In Section 11, we extend these methods to allow us to treat the ternary codes under consideration.

In [ l ] , [12], and [14] all self-dual ternary codes of lengths 20 or less are enumerated. In [lo] Leon, Pless, and Sloane show that there are only 2 inequivalent [24, 12,9] ternary self-dual codes. This classification required the complete enumeration of all 24 x 24 Hadamard matrices (see [8]). Such complete enumeration using these techniques seems infeasible for higher lengths. In the present paper, we give a partial enumeration of extremal codes for the next four lengths in which such codes exist. In particular, in Section 111 we enumerate the extremal self-dual ternary codes of lengths 28, 32, and 36 having a monomial automorphism of prime order r 2 5 and of length 40 having a monomial automorphism of prime order r > 5. Similar classifications for quaternary codes of lengths 18, 20, 22, 26, and 28 are given in [5] and [6]. It is interesting to note that the numbers of extremal ternary codes of lengths 28, 32, 36, and 40 are quite a bit larger than the numbers of extremal quaternary codes of lengths 18, 20, 22, 24, 26, and 28. (See also [9].) For example, there are 239 inequivalent [32, 16,9] ternary self-dual codes with an automorphism of order 5, far exceeding the numbers for any prime and any length considered in [5] or [6].

General references to coding theory are [ 1 11 and [ 15 3 .

11. DECOMPOSITION AND EQUIVALENCE

We summarize the theory developed in [4] and [SI on the decomposition of codes. Let F4 be the finite field of order q and characteristic p . Suppose r is relatively prime to p . Let R = F,[X] / (X ' - l ) , where X is an indeterminate. Suppose X' - 1 = I I~,om,(X) , where m , ( X ) is irreducible over F, and m o ( X ) = X - 1. Let Z, = ( ( X ' - l ) / m , ( X ) ) be the ideal of R gener- ated by ( X ' - l ) / m , ( X ) for 0 I; I g. By Lemma 1 of [4], R = Io d I, d . - . d Zg, Z, is a field for 0 I j 5 g , and Z,Zk = {0} if j # k. Let rPm,.: R + R be given by 7pu. u ( ~ ~ ~ ~ a l X f ) =

Manuscript received January 8, 1991. The author is with the Department of Mathematical Sciences, Loyola

University of Chicago, Chicago, IL 60626. IEEE Log Number 9107516.

1 : ~ d u f " X " ' where gcd( r , U) = 1. By Lemma 1 of [5], T ~ . , is a field automorphism of Io, permutes I,, . . , Zg , and if 7p-, I,) = Zkr rP-,

Let f? be a linear code over F, of length n and dimension k . The weight of a vector XEF," is the number of nonzero entries of x. The minimum distance d of V is the minimum nonzero weight of all codewords in W. V is called an [ n , k ] or [ n , k , d ] code. Let U be a permutation of the coordinates of F,". If XEF," has ith coordinate x , , define ( x u ) ; = x i o - ~ ; U is a permutation automor- phism of V if X U E W for all X E V. Assume U has only c r-cycles and f fixed points. Denote the r-cycles by Q , ; . ., 9 , and the fixed points by Q , , , , . . . , Q,,,. Let x 1 be the restriction of x to 0;. For 1 I i I e , x 1 *, can be viewed as an element a, + a , X + . . . +a,- ,X'- ' E R , where x u l o , is (a, + a , X + .. . + U , _ , X ' - ' ) X E R . Let C(u) = { X E %?I x u = x}, and for 1 5 j I g , E,(u) = { X E V I x I E Z , for 1 c: i I c and x I n, = 0 for c + 1 5 i c: c + f]. By Lemma 2 of [4], C ( u ) and E,(u) are U - invariant and V = C ( u ) e E, (u ) 8 . d ,!?,(U). Let Ej(u)* be E,(u) with the fixed points Q2,+,;. ., Q,,, deleted and the codewords viewed as c-tuples from If.

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

is a field isomorphism of Z, onto Zk.

n ( U , U) = c uiupm, (1)

i = 1

where U , UEF," with U = (U,;.., un) and U = ( u I ; . . , u n ) . De- fine Y L L = { u ~ F g " l ( u , u ) = 0 for all U E U } . V is left self- orthogonal under (1) if WE V and left self-dual if V =

analogously. If V ', define Y I = U L ; in this case if WE W , ie is self-orthogonal, and if V = V I , V is self-dual. (This is the case when considering ternary codes as p = q = 3 and p"' = 1 in (l).) The decomposition theorem of [SI is as follows.

,,IL 4 . Define V ', right self-orthogonality, and right self-duality = f?

Theorem I: Let s, t be nonnegative integers with s 5 m. Choose an integer U such that pSq'u = - 1 (mod r ) . Let (., .) be the form on RC given by

C

( x , y ) = x i y p . (2) i = 1

Let X be the permutation on 1, . . . , g where T ~ ~ - ~ , ,,(Zj) = ZWi) and let 0,;. ., 0, be the orbits of X. If V is a left self-dual [ n , n/2] code under (1) with permutation automorphism U , then C(u) is a left self-orthogonal [ n , (c + f ) /2] code under ( l ) , and for 1 I i 5 g , EWi,(u)* = ( 7 p m - l , u ( E j ( u ) * ) ) L under (2). Conversely, if C(u) is a left self-orthogonal [ n , (c + f ) / 2 ] code under (1) and if EWi)(u)* = (7pm-s,u(E, (u)* ) ) 'L under (2) for 1 5 i 5 g , then V is left self-dual under (1). In addition, c is even if 1 0'1 is odd for some J.

Define A,,(q) as the group of all n x n monomial matrices over F,, and define d x ( q ) as the semidirect product of Jn(q) ex- tended by Gal(F,), which is the Galois group of F, over Fp. A map T E A:( q) can be written as T = PD7 where P is a permuta- tion matrix (permutation part) , D is a diagonal matrix (diagonal part) , and r E Gal (F,). Linear codes V and %'I are equivalent whenever V' = V T for some T E A:( q). Define G( V) = { M E Ll , , (q) l V M = V } and G*( U) = { T E A,*(q)) U T = W}; G*( U ) is the automorphism group of U. Note that in the case of ternary codes, A x ( 3 ) = . l n ( 3 ) and G*( W ) = G( U). Also if V is a self-dual ternary code under (l), with pm = 1, V T for T E 4,(3) is also self-dual. The following result implies that in the ternary case, when considering monomial automorphisms of prime order

1396 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992

r # 2, we only need to examine permutation automorphisms. See also Theorem 2 of [3] for the quaternary case.

Lemma I: Let V be a linear code over F, with an automor- phism T = PDT of prime order r where r I ( q - 1) and r k I Gal (F,) 1 . Then there exists a code V' equivalent to V where

Proof: As r I 1 Gal (F,) 1 , T = 1 . The order of P must be r as r I I F," I where F," = F, - ( 0 ) . Order the coordinates so that P = (0, l ; . . , r - l ) ( r , r + 1 ; . * , 2 r - 1) . . . ((c - l ) r , ( c - Ilr + l ; . . , cr - 1). Let D = diag(d,;.., d + , ) where diEF4f( for 0 I i < n. Now ( P D ) ' = diag(e,;.., e n - l ) = I where e,,

1 = = di,+j for 0 I j < n - cr. As r k I F," 1 , dcr+j = 1 f o r O s j < n - c r . Let D, =diag(S, ; . . ,Sn- , )where6, ,= 1,

cr 5 m < n. As Il~,~dj,+k = 1 , D,PDD;' = P. Letting W' = VD;', we have V ' P = V'D,PDD; ' = VPDD;' = UD;' = V'. 0

We now summarize the results of [4], [5], [7], and [16] concern- ing the equivalence of two codes. For the remainder of this section, we assume V is an [n, k ] code over F, with a permutation automorphism U of prime order r where r I q, r I ( q - I) , and r k 1 Gal (F,) 1 . Assume U has c r-cycles and f fixed points where n = cr + f . Label the coordinates by (0 , 1,. . . , n - I } and order them so that U = U , u2 * . . U, where U,, , = ( s r , sr + 1, . . . , sr + r - 1) for 0 5 s < c. If i E Z , define i , = i(mod r ) , where 0 5 i, < r . For 1 I X < r , define f, as the permutation where ( s r + i)f , = sr + (A)), for 0 I s < c, 0 I i < r and xf, = x for cr I x < n. Let 8= {f,I 1 5 X < r } = Zp = Z, - { 0 } where Z, is the integers modulo r . Applying f , to replaces X by X' in each r-cycle of U . Let W = {uy' . . . U,"< 10 5 pf < r for 1 I t 5 c}. Application of an element of W to V simply cycles the entries of the r-cycles separately. Let C, be the symmetric group on { I , 2; . . , a}. If 4 E I,, define 4* as the permutation ((s - 1)r + i)4* = ((sb - l ) r + i ) for 1 5 s I c, 0 5 i < r and x4* = x for c r s x < n . Let E:= { d * I q b ~ C , } . If ~ E C , define # a s the permutation xq5' = x for 0 5 x < cr and (cr - 1 + i ) @ = cr - 1 + (id) for 1 5 i I f . Let X'' = {4' I 4 E Xf}. Application of an element of 1; to V permutes r-cycles; applying an element of C; to V permutes the fixed points. Let 9 = {diag ( a , , a , ; . . , a + , ) [ as, = as ,+ , = . . . = asr+r+l f o r 0 5 s < c}. Ap- plying elements of 9 to V scales each coordinate, with equal scaling on each coordinate of a given r-cycle. Finally, let Jt * =

The following result describes N * . Its proof is essentially the second paragraph of the proof of Theorem 2 of [ 5 ] , and is, therefore, omitted.

Lemma 2: If r k q , r I ( q - I ) , and r I (Gal (F,) 1 , then A'* = W C; I*, Q?.FGal (F,).

The next result is a combination of Theorems 2 and 3 and Lemma 5 of [5] and provides the essential tools we need to determine equivalence. Again its proof is omitted.

P E G ( U??.

- - e,,,, = ... - - e i r + r - , = I I i l h d i r + k = 1 for 0 I i < c and

6 . I ,+ , , = n' k = l d i r + k , and 6, = 1 for 0 5 i < c, 1 5 j < r , and

{ N E A;(q) 1 NUN-' E ( U ) } .

Theorem 2: Let V and V' have the same automorphism U of prime order r , where r k q, r I (q - I ) , r I lGal (F,) I . Then, we have the following.

a) If ( U ) is a Sylow r-subgroup of G*( U), then ,V and V' are equivalent, if and only if 0' = K N for some N E .A' *.

b) Suppose V = C(u ) Q E,(u) Q Q E,(u) and %" = C'( U ) CB E;( 0) CB * ' . CB E,(u) are the decompositions as de- scribed earlier. Let N E M * such that %" = VN. Then,

C ( u ) N = C'(u) and Ei(u)N = Elc i , (u ) where p is some permutation of 1 , . . . , g.

In testing for equivalence, we will use elements of &*. We describe the action of A'* induced on ,!?,(U)* and on the code C(u)@ that we now define. Note that I , = {a(l + X + . . . + X ' - ' ) I a EF } = F, and the isomorphism is given by a(1 + X + . . . +Xr-") + a r . (See Lemma 1 of [4].) Now define C ( u ) @ = { ( a ' r , . . . , a , r , a , , , , . . . , a , , , ) E

F ; + f ) ( a , C : , ; x i , - , a,Ci=,X ' , a , , , ; . . , a,,,) E C(u)}. r - 1

The groups W , C,, I:, 9, 9, and Gal (F,) induce natural actions on C(u)@ and E,(u)*. First, D and F a c t trivially on C(u)@; C, and permute fixed points and cycles, respectively, of C(u)@; 9, being constant on cycles, scales coordinates of C(u)@ and E,(u)*; and Gal(F,) retains its normal action on entries of C(u)%. On E,(u)*, C; acts trivially, and E*, permutes the coordinates; elements of W multiply coordinates of ,?,(U)* by powers u i , where U, is an element of I , of order r . Elements of .S and Gal (F,) induce permutations of the fields Ii for 1 5 i I g ; and if an element of F or Gal (F,) maps I , to I,, it acts as a field isomorphism. Let T E X r' C*,Q.FGal (F,) and let T I c(o)o be the induced action of T on C(u)@. From this discussion, the following is clear.

Lemma 3: Assume U E G*( U ) and U E G*( W?.

a) Assume C ( u ) = C'(u) and V T = W ' with T E

W Z'C*,9YGal(Fq) . Then T I c c , , o ~ G * ( C ( u ) @ ) . b) Define 4: D C; C*, 9 F G a l (F,) n G*( W) + G*(C(u)@)

by T@ = T 1 c(o)o. Then, 4 is a homomorphism with kernel W F . The image of 4 is contained in the subgroup of Az+,(q) consisting of maps whose permutation parts have no orbits with elements from both { 1,. . . , c} and { c + 1 , . . . , c

+f}. In testing for equivalence, we wish to apply Theorem 2a. In [4],

[5], [7], and [16] there are conditions under which ( U ) is a Sylow r-subgroup of G*( V); the next result gives another condition that allows us to apply Theorem 2a to all the cases we will examine in Section 111. First, let s/.,, , = { T E d;+,(q) I no orbit of the permu- tation part of T has elements from both { 1, * . . , c} and {c +

Lemma 4: Assume U E G*( V), r t q , r I ( q - l ) , and r i I Gal (F,) 1 . Assume all permutations of G( V ) of order r have exactly c r-cycles and f fixed points. Assume r I 1 G*(C(u)@) n x,, 1 . Then, ( U ) is a Sylow r-subgroup of G*( V ) .

is a Sylow r-subgroup of G*( U ) with U E %' and I 9i? I > r. Then, there exists T E g, T $ ( U ) such that T normalizes ( U ) . By Lemma 2, TE 'W C; C * , 9 F G a l (F,). As r t I G*(C(u)@) n x,, 1 , by Lemma 3 , T is in the kernel of 4 and hence, TE W Y . As Y normalizes %' and I Y 1 = r - 1, T E W . Therefore T = U:! . . . U,'. E C( V ) for some t ,,. . . , t , . As U = u1 . * . U,, then = U ; > - ' ] . . . U,'.-'!. As T , U E

G( U), Tu-'] E G( V ) . As T $ ( U ) and W is an elementary Abelian r-group, Tu-'! has order r with fewer than c r-cycles, a contradic- tion. 0

111. APPLICATIONS TO TERNARY CODES

l ; . . , c +f}}.

Proof: Assume

In this section, we find the extrema1 ternary self-dual codes of lengths n = 28, 32, and 36 which have a monomial automorphism of order r 2 5 and such codes of length n = 40 where r > 5 . By Lemma 1 we may assume the automorphism is a permutation. The element a, + a , X + . . . + a , - , X r - ' ~ F 3 [ X ] / ( X ' - 1) will be denoted a,a, . . . a,_ ,.

We first describe the fields ZI; * ., I , for the primes r I 19,

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992 1397

which are needed. Let I;” = I, - (0). If r = 5 , 7, 17, and 19, then g = 1 ; if r = 11, then g = 2; and if r = 13, then g = 4. Table I contains the information on the fields with 01 being the generator of I,#, p of I T , y of I F , and 6 of I?. The maps rpm-s,u from Theorem 1 are in Table I. Note that 71,1 is the identity maps X t o X - ’ . I n t h e c a s e r = l l o r r = 1 3 , ~ , , _ , ( a ) = P . In the case r = 13, T , , - , ( Y ) = 6. The inner product (., .) and order of I, are given in Table I. Note that a generator raised to the power ( 1 I, I - 1)/ r generates the subgroup corresponding to cyclic shifts; multiplication by powers of these elements to each cycle corresponds to the induced action of W acting on E,(u)*. The ternary inner product (1) is of course (U, U) = Cy= I u , ~ , . The term “left” in Theorem 1 can be omitted.

The codes we are examining are extremal with minimum distance d = 3[n/12] + 3; that is d = 9 for n = 28 and 32, and d = 12 for n = 36 and 40. Also n = Omod4 for extremal codes. (See [ l l , ch. 191.)

We find the values of r and the cycle structure of U for each n . Two facts eliminate many possibilities. First, by Theorem 1, if g = 1, then c must be even. As stated earlier g = 1 when r = 5 , 7, 17, and 19; g is also 1 when r = 29 and 31 and hence, r = 29 and 31 cannot occur as n 5 40. The second fact is Lemma 6 of [5] and Theorem 7 of [2] as follows.

Lemma 5: If f? has minimum distance d , then

a) if f s d - 1 , c r f ; b ) i f f r d , c + f r 2 d - 2 .

These two facts yield most of the following result.

Theorem 3: Let W be an extremal self-dual ternary code of length n = 28, 32, 36, or 40 with a permutation automorphism U

of prime order r 2 5 with c r-cycles and f fixed points. The following are the only possibilities:

a) n = 28, r = 7, c = 4, f = 0, b) n = 28, r = 13, c = 2, f = 2, c) n = 32, r = 5 , c = 6, f = 2, d) n = 32, r = 7, c = 4, f = 4, e) n = 36, r = 11, c = 3, f = 3, f ) n = 3 6 , r = 1 7 , c = 2 , f = 2 , g) n = 40, r = 5 , c = 8, f = 0, h) n = 40, r = 13, c = 3, f = 1, i) n = 40, r = 19, c = 2, f = 2.

Proof: The only possibilities not eliminated by the facts stated

1) When r = 5 and c = 2, E,(u)* is a [2,1] self-dual code over I , under the inner product of Table I. Thus, El(.)* contains a vector (a, ai) where 01 = 00012. But this vector corresponds to a nonzero vector of weight at most 7 in E,(u) .

2) When r = 7 and c = 2, E,(u)* is a [2, 11 self-dual code over I , under the inner product of Table I. As 1200000 E I , , E,(.)* contains a vector U = (1 + 2 X , p ( ~ ) ) for some p ( X ) € I I . p ( X ) = Cf=oajX’ must have weight 7 (i.e., a, # 0 for all i) as d 2 9. By examining I , , either five ai’s are 1 and two are 2, or five u,’s are 2 and two are 1. In either case U - U X is nonzero and has weight less than 9.

3) When either r = 11 and c = 1 or r = 13 and c = I , form the code 8 from e,“= ,E,(u) by dropping the fixed points. 8 is either an [ l l , 51 or [13,6] code. Neither code can have minimum distance 9 by the singleton bound ( [ l l , ch. 1 , Theorem 1 11 .)

4) When r = 5 , c = 4, and either f = 12 (when n = 32) or f = 20 (when n = 40), C(u)+ is a [16,8] or [24, 121 code,

prior to this theorem are easily decided as follows.

TABLE 1 FIELD INFORMATION

r Generator (U, U ) T ~ ~ - ~ ., Order

5 a=ooo12

c

U;$ i = I T I . I 81

7 cy = 1111200

r l , - I 243

13 cy = 0112111020210 i: U t v i 71. - 1 27

2 U t v i 11 a = 20111122011 i= I

p = 21102211110

I = I p = 0012020111211 y = 0012102211101 6 = 0101112201210

17 01 = 12111000000000000 2 u,u,~’ T , , ~ 316 i = I

C

19 a = 2111100000000000000 1 u i u ~ y 7 1 , 1 3’8 I = I

TABLE I1 GROUPS

28 7 4 0 28 13 2 2 32 5 6 2

32 7 4 4

36 11 3 3 36 17 2 2 40 13 3 1 40 19 2 2

48 8

288

128

48 8

12 8

5 )

respectively. Row reducing C(u)@ in the cycle coordinates first, the last two rows have at least 6 or 10 leading zeros, respectively, and supports entirely in the fixed point coordi- nates. Deleting these leading zeros gives a [lo, 21 or [14.2] code, respectively. It is straightforward to verify in each case that the minimum distance must be less than d . When r = 5 , c = f = 6 and n = 36, row reduce C(u)@ in the cycle coordinates first. The last row has its support on at most one cycle coordinate, leading to a vector in C(u) of weight less than 12. 0

With each case of Theorem 3, except case g), we found C(u)+, which turned out to be unique. If r 3 1 mod 3 or f = 0, C(u)@ is a self-dual ternary code by [4] and hence, found in [12]. The forms of C(U)+ are given in Theorems 4-7. We also computed G*(C(u)@) n q.,f; the results are in Table 11. In each case r I 1 G*(C(u)@) fl 9:. I . By Theorem 3 all permutations of order r for a given length have the same number of r-cycles and fixed points. Thus, by Lemma 4, ( U ) is a Sylow r-subgroup of G*( U). Hence by Theorem 2a, C‘ and W are equivalent, if and only if they are equivalent by an element in A *. By Lemma 2, JW* =

9.9 as Gal (F,) = I . Using elements of G*(C(u)@) n q,f, Theorem 2b, Lemma 3, and the induced action of W Y’c*,QLr on potential E,(u)*’s, a form for each E,(u)* is

found; we omit details here as similar techniques were used in [5] and [6]. By computer, equivalence classes under .A * were found, and a code in each class was tested for a nonzero low-weight vector.

C’

1398 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992

TABLE 111 VALUESOF @'FORSELECTED UWITHLY = 1 1 1 1 2 0 0 ~ ~ , [ x ] / ( x ~ - 1)

a cia a CYu U CYa U cy4

0 1 2 4 5 6 1 8 9

10 11 14 16 22

0222222 1111200 2001 102 0112110 0111210 20 102 10 2120112 0200202 1210101 0020001 1101222 0120012 21 12000 0112200

24 25 26 32 35 37 40 48 51 52 58 62 64 66

0002220 2012100 1102212 1121211 010121 1 221 1120 010 1022 2021001 0200121 1111011 0120102 0002001 202202 1 0021021

14 84 87 92

108 114 134 140 160 184 210 212 235 26 1

1100022 2 12222 1 2 122020 2011020 0011211 1002000 2oooo10 201 1 1 10 2 122002 2112111 022001 1 1001121 1110021 2221002

287 320 338 339 340 365 391 416 442 517 588 603 629 657

0012222 2020200 2121102 1111002 2 1222 12 2222100 2001222 2222022 2212110 200222 1 0012210 2002122 1112100 21 12201

The decomposition of F 3 [ X ] / ( X ' - 1) = e;=,Z, aided this search greatly. Again, more details are in [5] and [6]. In Theorems 4-7, the cycle coordinates are on the left. Also a + nX + . . . + ax'- will be denoted by a.

Theorem 4: Consider all [28, 14,9] self-dual ternary codes Y: with a monomial automorphism of prime order r 2 5. Then the automorphism can be assumed to be a permutation U of order r and the following are the only possibilities.

r = 7, c = 4, and f = 0; V = C(a) o E , ( u ) where gen

C(a) = [: :- and gen E, (a )* = 1. There i r e 14 inequivalent codes de-

termined by the following: (i, j , k , m ) : (1, 32, 32, 287), (1, 58, 58, 339), (1, 58, 84, 365), (1, 84, 84, 391), (2, 87, 87, 340), (4, 14, 14, 108), (4, 14, 66, 160), (4, 40, 40, 160), (4, 40, 92, 212), (7, 24, 24, 657), (10, 48, 48, 114), (10, 48, 74, 140), (16, 64, 64, 588), (26, 52, 52, 442). See Table 111 for values cia. r = 13, c = f = 2; V = Clu) o E I ( u ) e_E,(a) e E,(o) o

E,(a) where gen C(u) = . There are 5 in-

equivalent codes. The first has gen E,(o)* = Iff' 1 0 CY0 ' gen E,(a)* = {[OO]}, gen E,(a)* = [yoyo] , gen E4(u)* = [60813]. The remaining four have gen El(a)* = [aoa0] and gen E2(u)* = [poPl3]. One of these remaining four codes has gen E3(a)* = [yo 01 and gen &(U)* = [0 6'1. The other three have gen ,!?,(U)* = [yayJ] and gen ,!?,(a)* = [6°6'3-J] for j = 0, 1, and 4. Here CY' = 0101200221222, Po = 0222122002101, P I 3 = - P O , yo = 0212011222020, yl = 0 0 1 2 1 0 2 2 1 1 1 0 1 , y 4 = 2 0 2 0 0 2 1 2 0 1 1 2 2 , 6 0 = 0 0 2 0 2 2 2 1 1 0 2 1 2 , 6 9 = 0 1 2 1 0 0 1 0 1 1 1 2 2 , 6 1 2 = 2002022211021, 6 ' 3 = -60.

f fo 0 0 a' f f J f f k

0 0 a0 ffs 011 f f u

There are 239 inequivalent codes determined by (i, j , k , m , n, p , s, t , U) from Table IV. The value Cm represents 0. The values a', 0 5 a 5 16 are listed in Table V. Multiplica- tion of a' by a16 corresponds to a right cyclic shift of 1 position. This gives the remaining values of a' (e.g., =

( f f16)2f f9 = 00021). b) r = 7, c = f = 4; Y = C(u) E,(o) where

r l 0 0 0 1 1 0 0 i 0 1 0 0 0 0 1 1 and I I 0 0 0 1 0 0 1 2 0 0 1 0 1 2 0 0

gen C( a ) =

010 f f k am gen E, (a )* = [:

There are 16 inequivalent codes determined by the following: ( i , j , k , m ) : (0, 26, 26,416), (1, 6 ,6 ,235) , (1, 6, 32,261), (1 ,32,32,287) , ( 2 , 9 , 9 , 184), (2 ,9 ,35,210) , (4, 14, 14, 108), (4, 14,40, 134), (4 ,40,40, 160), (5,25,25,577), (5 ,25,51, 603), (5, 51, 51, 629), (8, 11, 37, 320), (10, 22, 22, 62), (10,48,48, 114). (26,0,0,388) . See Table 111 for values au.

Theorem 6: Consider all [36, 18, 121 self-dual ternary codes V with a monomial automorphism of prime order r 2 5. Then the automorphism can be assumed to be a permutation U of order r = 17 with c = f = 2 where V = C(o) e E, (a ) with gen C(u) =

[ A 71 and gen El(a)* = [ c Y O ~ ' ~ ~ ] . Here, 01' =

21111111111111111, p = 013280 and p193 = 11122111212202212. This code is the Pless symmetry code of [13].

Note that case e) of Theorem 3 does not yield an extrema1 code. In that case,

Theorem 5: Consider all [32, 16,9] self-dual ternary codes V with a monomial automorphism of prime order r 2 5. Then the automorphism can be assumed to be a permutation U of order r and the following are the only possibilities.

%'= C(o) o E , ( o ) o &(a) and

1 0 0 1 0 0 o 1 o o 1 o , 0 0 1 0 0 1 1

a) r = 5, c = 6, and f = 2; V = C(a) o E, (o ) where gen E ' ( u ) * = [ aOa'a~],

gen E2(u)* = 0 1 0 0 1 0 2 2 1 0 0 0 1 0 1 1

0 0 1 0 0 1 1 2 0 0 0 1 0 1 2 1

There are 6 inequivalent codes with ( i , j ) = (O,O), (0, l) , (0,2), (1,2), (1,3), and (1,5). All have weight 9 codewords.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992 1399

TABLE IV THEOREM 5 WITH r = 5, c = 6, AND f = 2

(m, n, P, s, I , U): (3,0,22, 1,58, O), (7, 1,23,2,6,28), (1,2,24,3,24,46), (1,2,24, 11,32,54), (5,5,27, 14,44,66), (13,5,27,6,28,50), (2,6,28,7, 1,23), (2,6,28, 15,9,31), ( IO, 6,28,7,73, 15), (7,9,31,2, 14,36), (1,10,32,3,32,54), (1,10,32, 11,40,62), ( 5 , 13,35,6,44,66), ( 5 , 13,35, 14,52,74), (2,14,36,7,9,31), (2, 14,36, 15, 17,39)

( i , j , k ) : ( - =,2,7)

(m, n , p , s, I , U): (5, 1,76,6,32,27), (5 , 1,76, 14,40,35), (7,5,0,2, 10,5)

( i J , k ) : (0, 1,2)

(m, n, p , s, 1, U): (1,0,42,2,42,61), (1,0,42, 10,50,69), (1,12,53,2,53,34), (1, 13, 12,2,12,78), (1,13, 12,10,20,6), (1, 17,30, 10,38,28), (1,21,4,2,4,30), (1,22,73, 10, 1,72), (1,34,40, 10,48, 17), (1,50, 16, 10,24,57), (1,57,6,2,6,68), (1,57,6, 10,14,76), (2,4,30, 1,21,4), (2, 12,78,9,21,20), (2, 16,49, 1,50, 16), (2, 16,49,9,58,24), (2,30,20, 1, 17,30), (2,30,20,9,25,38), (2,34,37, 1,64, 34). (2,40,9, 1,34,40), (2,40,9,9,42,48), (2,42,61, 1,0,42), (2,42,61,9,8,50), (2,53,34,9,20,61), (2,73,64,1,22,73), (3,6,24,6,33,25), (3,6,24, 14,41,33), (3, 10,3,6,64,59), (3,21,53,6,52, lo), (3,21,53, 14,60, 18), (3,58,67,6,40,67), (3,63,20,6,23,62), (3,77,37,6, 12,26), (3,77,37, 14,20,34), (4,0,52,4,34,78), (4,9,5,4,46,58), (4,26, 14,4,40,20), (4,26, 14, 12,48,28), (4,34,78,4,0,52), (4,39,67,4,71,75), (4,39,67, 12,79,3), (4,40,20,4,26, 14), (4,40,20, 12,34,22), (4,41,77, 12,62,2), (4,54,74,4,41,77), (4,54,74, 12,49,5), (4,71,75,4,39,67), (4,71,75,12,47,75), ( 5 , 13,11, 15,68,75), (5 , 18,42,7, 16,5), (5,28,73,7,69,56), (5,31,56,7,79,33), (5,31,56, 15,7,41), (5,34,2,7,0,53), (5,34,2, 15,8,61), (5,44,41, 15,5,40), (5,49,28,7,22,6), (5,49,28, 15,30, 14), (5,69,43,7,76,51), (5,71,48,15, 15,73), (5,73,36,7,54,62), (6, 1, 1,3,22, 16), (6, 1,1,11,30,24), (6,23,62,3,63,20), (6,33,25,3,6,24), (6,33,25, 11, 14,32), (6,40,67,3,58,67), (6,52, 10,11,29,61), (6,64,59, 11, 18, 1 l) , (6,71, 14,3,55,60), (6,71, 14, 11,63,68), (7,0,53,5,34,2), (7,7,65,5,71,48), (7,22,6,5,49,28), (7,54,62,13,1,44),(7,60,67,5,13,11),(7,69,56,5,28,73), (7,69,56, 13,36, l), (7,76,51, 13,77,51), (7,79,33,5,31,56), (9, 8,50,2,42,61), (9, 8,50, 10,50,69), (9,21,20,2, 12,78), (9,21,20, 10,20,6), (9,25,38,2,30,20), (9,25,38, 10,38,28), (9,42,48, IO, 48, 17), (9,58,24, 10,24,57), (9,65, 14, 10, 14,76), (9,72,42, 10,42,45), (IO, 1,72, 1,22,73), (10,1,72,9,30, l), (10, 12,38, 1,21,4), (10, 14,76, 1,57,6), (10,24,57, 1,50, 16), ( I O , 24,57,9,58,24), ( IO, 38.28, 1, 17,30), (10,42,45,1,64,34), (10,48, 17,1,34,40), (10,50,69, 1,0,42), ( IO, 61,42, 1. 12,53), (10,61,42,9,20,61), (11, 14,32, 14,41,33), (11,30,24, 14,9,9), (12, 17, 13,4,46,58), (12,34,22, 12,48,28), (12,42,6, 12,8,60), (12,47,75,4,71,75), (12,48,28, 12,34,22), (12,54,66, 12, 17,13), (12,62,2,4,41,77), (12,62,2, 12,49,5), (13, 1,44,7,54,62), (13, 1,44, 15,62,70), (13,26,50,7, 16,5), (13,36,1,7,69,56), (13,39,64,7,79,33), (13,39,64, 15,7,41), (13,42, IO, 15,8,61), (13,57,36,7,22,6), (13,57,36, 15,30, 14), (13,77,51,7,76,51), (13,79,56, 15,15,73), (14,9,9,11,30,24), (15,4,59, 13,77,51), (15,5,40,5,44,41)

( j , j , k ) : (0,1,10) (m, n, p , s, t , U): (1,0,50, 10,50,77), (1,57, 14,10, 14,4), (2,4,38,1,21,12), (2,4,38,9,29,20), (2, 12,6, 1, 13,20), (2, 16,57, 1,50,24), (2, 16,57,9,58,32), (2,34,45, 1,64,42), (2,40, 17, 1,34,48), (2,53,42, 1, 12,61), (2,53,42,9,20,69), (2,73,72, 1,22, I ) , (3,63,28, 14,31,78), (4,34,6,4,0,60), (4,40,28,4,26,22), (4,71,3,4,39,75), ( 5 , 13, 19,7,60,75), (5,28,1,7,69,64), (5,44,49,15,5,48), (5,71,56,7,7,73), (5,71,56, 15, 15, I), (6,23,70, 11,71,36), (9,21,28, 10,20, 14), (9,25,46, 10,38,36), (12,34,30, 12,48,36), (12,62, 10,12,49, 13), (13,1,52,15,62,78), (13,39,72,7,79,41), (13,57,44,7,22, 14)

TABLE IV-(Continued)

( i , j , k ) : (0,4,4)

(m, n, P, s, t , U): (4,9,21,4,21,9), (4,9,21, 12,29,17), (4,39,35,4,35,39), (4,58,78,4,78,58), (4,58,78,12,6,66), (5,7,64,7,75,68), (5,7,64,15,3,76), (5,20,41,7,16,5), (5,20,41,15,24, 13), (5,24,71,7,44,27), (5,24,71, 15,52,35), (5,41,20,7,5, 16), (5,41,20, 15, 13,24)

( i J ,k ) : (0,4,12) (m, n, p , s, t, U): (4,9,29.4,21, 17), (4, 11,55,4,47, 19), (4,58,6,4,78,66), (4,78,66,4,58,6), (5,7,72,7,75,76), (5,20,49,7, 16, 13), (5,20,49, 15,24,21), (5,24,79,15,52,43), (5,41,28, 15, 13,32), (7,5,24,5,41,28), (7, 16, 13,5,20,49), (7, 16, 13, 13,28,57), (7,44,35,5,24,79)

( i J , k ) : (0,5,7) (m, n, p, s, t, U): (7, 13, 16,5,27, 13), (7, 13, 16, 13,35,21), (7,43,68,5,69,43), (15,21,24,5,27, 131, (15,29,40,5,3,21), (15,59,28,5,61,51)

( i , j , k ) : (1 ,O . 2)

(m, n, p, s, t, U): (2,53,50,8, 15,61), (4,45, 1, 13,23,48), (4,78,18, 13,19,53), (9,45,68,6,71,70), (9,45.68, 14,79,78), (9,62,1,14,11,32), (12,53,9,5,15,40)

( i J , k ) : (1,4,5)

(m, n , P , s, t, U): (4,8,76,5,76,65), (4,8,76, 13,4,73), (4, 16,60,5,60,33), (4,21,49,5,49,36), (5,49,36,4,21,49), (13,4,73,4,8,76),(13,57,44,4,21,49)

( i J , k ) : (1,5,4)

(m, n, p, s, t, U): (1,72,42,6,31,54), (1,77,60,6, 11,32), (1,77,60, 14,19,40), (2,40, 1,8,23,53), (4,2,22,5,66,2), (4,2,22,13,74, lo), (4,34,62,5,42,34), (4,40,28,5,47,40), (4,40,28, 13,55,48), (4,56,36,5,31,56), (4,59,23, 13,21,67), (9,5,68,6,11,32),(12,42,70,5,42,34),(12,64,44,5,31,56)

( i , j , k ) : (2,4,6)

(m. n. LJ. s. 1. U): (4.22.42.6. 15.30)

TABLE V VALUES OF aa FOR SELECTED a WITH a = 00012 E~3[x]/(x5 - I )

a CYa a cia a aa a f f a

- 00 00000 4 21120 9 02100 13 21102 0 21111 5 01122 10 22002 14 02202 1 00012 6 01011 11 10200 15 01212 2 01110 7 21021 12 21012 16 12111 3 00201 8 22212

Theorem 7: Consider all [40,20, 121 self-dual ternary codes V with a monomial automorphism of prime order r > 5. Then the automorphism can be assumed to be a permutation U of order r and the following are the only possibilities.

r = 13, c = 3, and f = 1; g= C(u) E 3 ( u ) @ E4(u) where gen C ( u ) =

. There are four nonequivalent codes

determined by the following ( i , j ) : (2,2), (4,0), (4, l) , and (5,22). Here, (YO = 0101200221222, Po = 0222122002101,

1

1400 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992

TABLE VI COMPUTER RESULTS

Number of Number of Inequivalent Extrema1

n r Codes Codes

28 28 32 32 36 36 40 40

7 13 5 7

11 17 13 19

31 5

310 18 6 13 165 31

14 5

239 16 0 1 4 11

p’3 = - P O , yo = 0212011222020, yl = 0012102211101, y2

= 2002120112220, y 4 = 2020021201122, y5 = 1 1 0 1 0 0 1 2 1 0 2 2 1 , y Z 2 = 0 1 1 2 2 2 0 2 0 0 2 1 2 , 6 0 = 0 0 2 0 2 2 2 1 1 0 2 1 2 , 8 1 1 = 2 1 0 0 1 0 1 1 1 2 2 0 1 , 61Z = 2002022211021, 613 = -60 , 616 = 0222110212002, 6 l7 = 1112201210010, and 6’’ = 0212002022211. r = j 9 , c = f = 2; F= C(u) @ E,(u) where gen C ( u )

= 1; :I and gen El(a)* ~ = [ a O p i ] with a’ =

0222222222222222222 and u9841. There are 1 1 ineauivalent codes for the values i = 1, 5; 19, 29, 49, 59, 65, 83, $7, 173, and 259. Here pLI = 0201020101101001020, p5 = 0012112110101100120, PI9 = 0210211010122020210, J’ = 2021201211222012021, up’ = 2120021222101100010, p” = 2212100022021121212, yp’ = 1211001220010221002, p*’ = 0001202120121201102, p’’ = 1210221102001112112, $73

= 2 1 1 2 1 2 0 2 2 2 2 1 1 0 0 2 0 2 1 , a n d p 2 5 9 = 0220020022220202000.

IV. CONCLUSION

We conclude with a few remarks. First, the case r = 5 when n = 40 (case g) of Theorem 3) was not done. In that case, C(o)@ is E4 @ E4 where E4 is the [4 ,2 ,3] tetracode (see [12]); but the number of possible codes for El(a)* is extremely large. However, 5 ! 1 G*(C(a)@) n Y8,0 I in that case as well, and hence, the techniques of Section I1 on equivalence can be applied. Second, the equivalence or inequivalence of two extremal codes of length n constructed from two different values of r is still an open problem. Because the general question of when two codes are equivalent is so difficult, the power of results such as Theorem 2 becomes clear when by computer, it was relatively easy and quick to decide that the 239 [32, 16,9] codes with r = 5 are inequivalent. Third, the author was very surprised that the number of extremal codes found was so large. Also the high percentage of codes examined that turned out to be extremal was a surprise. This is illustrated by Table VI. In this table, “Number of inequivalent codes” refers to the number of equivalence classes of codes that were examined by computer; here the equivalence classes were those determined as if Theorem 2 held. We checked the general forms that were given in Theorems 4-7. A similar table appears in [5] for quaternary codes, and we see by comparing these two tables that the percentage of extremal codes in the ternary case is much higher. Codes of length 40, with r = 5, and 44, with r >- 5, might be interesting to examine if it becomes computationally feasible. Fourth, one might ask if the Pless symmetry code of length 36 is the unique extremal code of that length. Finally, the programming required for this paper was done on an AT & T 6300 in Pascal.

REFERENCES 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 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,” DiscreteMath., vol. 38, pp. 143-156, 1982. - , “Monomials of orders 7 and 11 cannot be in the group of a (24, 12, 10) self-dual quaternary code,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 137-140, Jan. 1983. 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, May 1988. -, “On extremal self-dual quaternary codes of lengths 18 to 28, I ,” IEEE Trans. Inform. Theory, vol. 36, pp. 651-660, May 1990. - , “On extremal self-dual quaternary codes of lengths 18 to 28, 11,“ IEEE Trans. Inform. Theory, vol. 37, pp. 1206-1216, July 1991. - , “On the equivalence of codes and codes with an automorphism having two cycles,” Discrete Math., vol. 83, 265-283, 1990. N. Ito, J. S. Leon, and J. Q. Longyear, “Classification of 3-(24,12,5) designs and 24-dimensional Hadamard matrices, ” J . Combinat. Theory A , vol. 31, pp. 66-93, 1981. C. W. H. Lam and V. Pless, “There is no (24, 12, 10) self-dual quaternary code,” IEEE Trans. Inform. Theory, vol. 36, pp. 1153- 1156, Sept. 1990. J . S. Leon, V. Pless, and N. J. A. Sloane, “On ternary self-dual codes of length 24,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 176-180, 1981. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes. Amsterdam: North-Holland, 1977. C. L. Mallows, V. Pless, and N. J. A. Sloane, “Self-dual codes over

V. Pless. “Symmetry codes over GF(3) and new five-designs,’’ J . Combinat. Theory, vol. 12, pp. 119-142, 1972. V . Pless, N. I. A. Sloane, and H. N. Ward, “Ternary codes of minimum weight 6 and the classification of the self-dual codes of length 20,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 305-316, 1980. V. Pless, Introduction to the Theory of Error-correcting Codes, second ed. V. Y. Yorgov, “A method for constructing inequivalent self-dual codes with applications to length 56,” IEEE Trans. Inform. The- ory, vol. IT-33, pp. 77-82, Jan. 1987.

GF(3),” SIAM J . Appl. Math., vol. 31, pp. 649-666, 1976.

New York: John Wiley, 1989.

A Fast On-Line Adaptive Code

Boris Ya. Ryabko

Abstract-There are two classes of data compression algorithms. One class has redundancy log log n + 0(1), where n is the alphabet size, and an encoding time O(log2n), n + 00. The other has redundancy O(1) and an encoding time O(n). A code is presented combining advantages of both classes of compression methods: its redundancy is O(1) and the encoding and decoding time is O(log2 n ) per letter, which is close to the lower bound O(1og n).

Index Terms-On-line adaptive coding, Huffman code, encoding, book-stack method.

INTRODUCTION

The first adaptive on-line code was apparently proposed in 1980 [14], and then rediscovered in the papers of Bentley, Sleater, Tarjan

Manuscript received January 17, 1990. The author is with Applied Mathematics and Cybernetics, Novosibirsk

Institute of Communication, Kirov Street 86, Novosikizsk-125, Russian Federation.

IEEE Log Number 9108027.

0018-9448/92$03.00 0 1992 IEEE