[3] E. Bannai, E. Bannai, M. Ozeki, and S. Teranishi, “Rings of simulta-neous invariants for the MacWilliams–Gleason group,” Europ. J. Comb., vol. 20, pp. 619–627, 1999.
[4] G. Batut, D. Bernardi, H. Cohen, and M. Olivier, “PARI-GP,” a computer algebra system.
[5] R. A. Brualdi and V. Pless, “Weight enumerators of self-dual codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1222–1225, 1991.
[6] R. A. Brualdi, S. Litsyn, and V. Pless, “Covering radius,” in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: North-Holland, 1998.
[7] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes. Amsterdam, The Netherlands: North-Holland, 1997. [8] J. H. Conway and N. J. A. Sloane, “A new upper bound on the minimal
distance of self-dual codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1319–1333, 1990.
[9] , Sphere Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, 1998.
[10] J. H. Conway, V. Pless, and N. J. A. Sloane, “The binary self-dual codes of length up to 32: A revised enumeration,” J. Comb. Theory Ser. A, vol. 60, pp. 183–195, 1992.
[11] P. Delsarte, “Four fundamental parameters of a code and their combina-torial significance,” Inform. Contr., vol. 23, pp. 407–438, 1973. [12] , “An algebraic approach to the association schemes of coding
theory,” Philips Res. Repts. Suppl., vol. 10, 1973.
[13] A. M. Gleason, “Weight polynomials of self-dual codes and the MacWilliams identities,” in Actes Congres Internl. de Mathematique 1970. Paris, France: 3 Gauthier-Villars, 1971, pp. 211–215. [14] M. Harada and M. Ozeki, “Extremal self-dual codes with the smallest
covering radius,” Discr. Math., pp. 271–281, 2000.
[15] F. Klein, Vorlesungen über das Ikosaeder. Leipzig, Germany: Taubner, 1884. Reprinted: Bawl, Switzerland: Birkhäuser, 1993.
[16] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [17] M. Ozeki, “Hadamard matrices and doubly even error correcting codes,”
J. Comb. Theory Ser. A, vol. 44, pp. 274–287, 1987.
[18] , “Determination of the ring of simultaneous invariants for a group associated with MacWilliams identity (an intermediate report of a joint work with E. Bannai),” in Meeting on Algebraic Combinatorics, Suuri Kaiseki Kenkyuusho Koukyuuroku, to be published.
[19] , “On the notion of Jacobi polynomilas for codes,” in Math. Proc. Cambridge Philos. Soc., vol. 121, 1997, pp. 15–30.
[20] , “On covering radii and coset weight distributions of extremal bi-nary self-dual codes of length 40,” Theor. Comp. Sci., vol. 235, pp. 283–308, 2000.
[21] , “Determination of covering radii and coset weight distributions of doubly even extremal binary self-dual codes of length 56,” IEEE Trans. Inform. Theory, vol. 46, pp. 2359–2372, Nov. 2000.
[22] , “Notes on the shadow process in coding theory,” Discr. Math., to be published.
[23] V. Pless, “A classification of self-orthogonal codes over GF(2),” Discr. Math., vol. 3, pp. 209–246, 1972.
[24] V. Pless and N. J. A. Sloane, “On the classification and enumeration of self-dual codes,” J. Comb. Theory Ser. A, vol. 18, pp. 313–335, 1975. [25] V. Pless, An Introduction to the Theory of Error-Correcting
Codes. New York: Wiley, 1982.
[26] E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: North-Holland, 1998.
[27] E. M. Rains, “Shadow bounds for self-dual codes,” IEEE Trans. Inform. Theory, vol. 44, pp. 134–139, Jan. 1998.
[28] I. Schur, Vorlesungen über Invariantentheorie. Berlin, Germany: Springer, 1968.
[29] G. C. Shephard and J. A. Todd, “Finite unitary reflection groups,” Canad. J. Math., vol. 5, pp. 364–383, 1953.
[30] N. J. A. Sloane, “Error-correcting codes and invariant theory: New ap-plications of a nineteenth-century technique,” Amer. Math. Monthly, vol. 84, pp. 82–107, 1977.
[31] , “Self-dual codes and lattices,” in Relations Between Combina-torics and Other Parts of Mathematics, Proc. Symp. Pure Mathematics, vol. 34, 1979, pp. 273–308.
[32] V. D. Tonchev, “Hadamard-type block designs and self-dual codes,” Probl. Inform. Transm., vol. 19, pp. 270–274, 1983.
[33] H. Chen and J. T. Coffey, “Trellis structure and higher weights of ex-tremal self-dual codes,” Des., Codes Cryptogr., vol. 24, pp. 15–36, 2001.
On the Doubly-Even Self-Dual Codes of Length
Radinka Dontcheva
Abstract—We prove that23 11 and 7 do not divide the order of the
automorphism group of a binary [96 48 20] doubly-even self-dual code. We construct 25 new inequivalent binary[96 48 16] doubly-even self-dual codes via an automorphism of order23.
Index Terms—Automorphisms, doubly-even self-dual codes, weight
enu-merators.
I. INTRODUCTION
An[n; k] linear code C over the binary field F2is ak-dimensional subspace ofF2n, where F2n is then-dimensional vector space. The
weight of a vector inF2nis the number of its nonzero coordinates. The
minimum distanced of C is the minimum weight of its nonzero
code-words andC is called an [n; k; d] code. For every
u = (u1; u2; . . . ; un)
v = (v1; v2; . . . ; vn) 2 F2n
u:v = u1:v1+ u2:v2+ 1 1 1 + un:vn2 F2
defines the inner product inF2n.
The dual code of codeC is C?= fv 2 F2n: u:v = 0 for all u 2 Cg. IfC C?,C is called self-orthogonal and if C = C?,C is self-dual. A doubly-even code is a binary code in which the weight of every vector is divisible by four. Self-dual doubly-even code exists only when its lengthn is a multiple of eight and its minimum distance d satisfies
d 4[n
24] + 4 (see [5]). If d = 4[24n] + 4, C is called extremal.
For a permutation of n elements and v = (v1; v2; . . . ; vn) 2 F2n
we define
v = (v1 ; v2 ; . . . ; vn ):
The binary codeC and its image C are called equivalent. If C = C then the permutation is an automorphism of C. The set of all auto-morphisms of the codeC forms the automorphism group of the code C. A list of the possible weight enumerators for extremal self-dual codes of length from72 to 100 was given by Dougherty, Gulliver, and Harada in [2]. The existence of a binary[96; 48; 20] self-dual doubly-even code is still unknown.
In [11], Shterev, Yorgov, and Ziapkov proved that a binary
[96; 48; 20] self-dual doubly-even code cannot have automorphisms
of order47 and 31. Bouyuklieva shows that the same code cannot have an automorphism of order2 with f-fixed points for f > 0 (see [1]).
The largest minimum distance for a known [96; 48] self-dual doubly-even code is16. The first example of a [96; 48; 16] code was found by Feit in [3].
We use the method for constructing binary self-dual codes with an automorphism of an odd prime order developed by Huffman and Yorgov in [4], [9] and [10]. In Section II, for completeness, we
Manuscript received October 25, 2000; revised February 3, 2001. This work was supported in part by the Bulgarian National Sscience Foundation under Grant MM-901/99.
The author is with the Faculty of Information Technology and Systems, Delft University of Technology 2628 CD Delft, The Netherlands, on leave from the University of Shumen, Bulgaria.
Communicated by P. Solé, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(02)00306-1.
describe the method and prove that a binary [96; 48; 20] self-dual doubly-even code cannot have an automorphism of order23; 11; or
7. In Section III, we present 25 new binary [96; 48; 16] self-dual
doubly-even codes.
II. DESCRIPTION OF THEMETHOD ANDGENERALRESULTS
LetC be a binary [n; n=2; d] self-dual doubly-even code with an automorphism of an odd prime order p with c cycles and f fixed points in its decomposition. In short, we say that is of type p-(c; f). Let1; 2; . . . ; cbe the cycles andc+1; c+2; . . . ; c+fbe the fixed points of. Denote F(C) = fv 2 Cjv = vg and
E(C) = fv 2 Cjwt(vji) 0(mod 2); i = 1; . . . ; c + fg
wherevjiis the restriction ofv on i.
Lemma 1 [4]: If the codeC is self-dual then C = F(C) 8 E(C)
(8 denotes the internal direct sum) and dimF E(C) = (p01)c2 . If2 is a primitive root(mod p) then c is even.
By Lemma 1, a generator matrix of the codeC can be represented in the form
cycles xed points
gen(C) = A 0
X Y
g gen(E(C))
g gen(F(C)):
Denote the map: F(C) ! F2c+f, defined by(vji) = vj for some j 2 i; i = 1; 2; . . . ; c + f. It is known that (F(C)) is
a binary[c + f; c+f2 ] self-dual code [4]. Every vector of length p is identified by a polynomial in the factor-ringF2[x]=(xp0 1), namely,
(v0; v1; . . . ; vp01) corresponds to v0+v1x+1 1 1+vp01xp01. LetP be the set of even-weight polynomials inF2[x]=(xp0 1). It is known thatP is a cyclic code of length p generated by x 0 1.
Denote by E(C)3 the code E(C) with the last f coordinates
deleted. Forv 2 E(C)3we can consider each
vji= (v0; v1; . . . ; vp01)
as a polynomial
'(vji)(x) = v0+ v1x + 1 1 1 + vp01xp01
inP , i = 1; 2; . . . ; c. In this way we define the map ': E(C)3! Pc:
It is known [4] that'(E(C)3) is a submodule of the P -module Pc
and for eachu; v 2 '(E(C)3) it holds (see [9])
u1(x)v1(x01) + u2(x)v2(x01) + 1 1 1 + uc(x)vc(x01) = 0: (1)
Let
xp0 1 = (x 0 1)h1(x)h2(x) 1 1 1 hs(x)
wherehj(x) is an irreducible polynomial in P for j = 1; 2; . . . ; s.
Thus,P = I18I281 1 18IswhereIj = hx 01h (x)i for j = 1; 2; . . . ; s is an irreducible cyclic code and
Mj = fu 2 '(E(C)3)jui2 Ij; i = 1; 2; . . . ; sg
is a code over the fieldIjforj = 1; 2; . . . ; s. It is proved in [9] that
'(E(C)3) = M18 M28 1 1 1 8 Msanddim '(E(C)3) =cs2.
Lemma 2 [9]: Suppose thatC has an automorphism of type p-(c; f).
The following transformations lead to an equivalent code: i) a permutation of the firstc cycles of C;
ii) a permutation of the lastf coordinates of C;
iii) a multiplication of thejth coordinate of '(E(C)3) by xt , wheretjis an integer,1 tj p 0 1 for j = 1; 2; . . . ; c;
iv) a substitutionx ! xj forj = 1; 2; . . . ; p 0 1 in '(E(C)3).
LetC be a binary [96; 48; 20] self-dual doubly-even code with an automorphism of type p-(c; f). The only possibilities for p-(c; f) are as follows:23-(4; 4), 11-(8; 8), 7-(12; 12), 7-(13; 5), 5-(16; 16),
5-(18; 6), 3-(24; 24), 3-(26; 18), 3-(30; 6), and 3-(32; 0) [11].
The weight enumerator of the codeC is uniquely determined in [2]
W (y) = 1 + 3217056y20+ 369844880y24+ 18642839520y28
+422069980215y32+ 4552866656416y36+ 1 1 1 : (2)
In what follows, we prove that the maximal possible odd prime order of the automorphism ofC is 5.
Proposition 1: A self-dual doubly-even[96; 48; 20] code C cannot
have an automorphism of order23.
Proof: SupposeC has an automorphism of order 23. Then,
is of type23-(4; 4) and (F(C)) is a binary [8; 4] self-dual code.
Such codes are onlyA8orC24(see [6]) with generator matrices
gen(A8) = 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 1 0 and gen(C4 2) = 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 :
Let(F(C)) = A8. Since the automorphism group ofA8is 3-tran-sitive, without loss of generality (w.l.o.g.) we may assume that the last three positions ingen(A8) are fixed under . The forth fixed point can
be selected among the first five columns. It is easy to see that in any case the codeF(C) will contain a vector of weight 4 or 26. This
elim-inatesA8.
If (F(C)) is C24 then we may choose the generator matrix of
F(C) in the form
cycles xed points gen(F(C)) = a 1 0 0 0 a 0 1 0 0 a 0 0 1 0 a 0 0 0 1 (3)
wherea is the all-one vector of length 23 and nonindicated entries are equal to zero.
Sincex230 1 = (x 0 1)h1(x)h2(x), where
h1(x) = x11+ x10+ x6+ x5+ x4+ x2+ 1
are irreducible polynomials overF2, it follows that P = I1 8 I2,
Ij = hx 01h (x)i for j = 1; 2. Hence '(E(C)3) = M18 M2 and
dim '(E(C)3) = 4.
Denote
j(x) = x 230 1
hj(x) ; forj = 1; 2:
ThenIj = f0; kj(x)jk = 0; 1; . . . ; 2110 2g for j = 1; 2. The
multiplicative order ofj(x) is 2110 1 = 23 2 89 and we can write
j(x) = x j(x), where the multiplicative order of j(x) is 89 for
j = 1; 2. The idempotents of I1andI2are
e1(x) = x22+ x21+ x20+ x19+ x17+ x15+ x14
+x11+ x10+ x7+ x5+ 1
ande2(x) = e(x) 0 e1(x), where e(x) = x22+ x21+ 1 1 1 + x is the
identity ofP .
We havedim M1+ dim M2 = dim '(E(C)3) = 4. Applying
transformations i), iii), and a multiplication with a nonzero element ofI1 we obtain the generator matrix ofM1in the form[IjG], where
I is the identity matrix, G is a matrix with elements from I1. Thus,
dim M1= 2 or 3 and the generator matrix of M1can be chosen in the form L1= e1 0 0 t1 (x) 0 e1 0 t1 (x) 0 0 e1 t1 (x) wheretl = 0; 1; . . . ; 88 for l = 1; 2; 3 L2= e1 0 0 t1(x) 0 e1 t1 (x) xk t1 (x) or L3= e1 0 t1 (x) t1(x) 0 e1 t1 (x) xk t1 (x)
wheretl = 0; 1; . . . ; 88 for l = 1; 2; 3; 4 and k = 0; 1; . . . ; 22.
Denote
1(x) = x20+ x17+ x15+ x14+ x13+ x12
+x11+ x10+ x7+ x3+ x + 1:
Applying the substitution x ! x2 to the row vector
(e1(x); 0; 3; t1 (x)) for t1 = 0; 1; . . . ; 88 we can limit its
choice to the set of vectors:
f(e1(x); 0; 3; e1(x)); (e1(x); 0; 3; 1(x)); (e1(x); 0; 3; 31(x));
(e1(x); 0; 3; 51(x)); (e1(x); 0; 3; 91(x)); (e1(x); 0; 3; 111 (x));
(e1(x); 0; 3; 131 (x)); (e1(x); 0; 3; 191 (x));
(e1(x); 0; 3; 331 (x))g
where3 is any element of I1. By transformation i) and a multiplication with a nonzero element ofI1we can reduce the above set to the fol-lowing set
f(e1(x); 0; 3; e1(x)); (e1(x); 0; 3; 1(x)); (e1(x); 0; 3; 31(x));
(e1(x); 0; 3; 51(x)); (e1(x); 0; 3; 131 (x))g:
Therefore, it is sufficient to consider only generator matrices ofM1in the form L0 1= e1 0 0 t1 (x) 0 e1 0 t1 (x) 0 0 e1 t1 (x) (4) wheret1= 0; 1; 3; 5; 13, tl= 0; 1; 1 1 1 ; 88 for l = 2; 3 L0 2= e1 0 0 t1 (x) 0 e1 t1 (x) xk t1 (x) (5) or L3= e1 0 t1 (x) t1 (x) 0 e1 t1 (x) xk t1 (x) (6)
wheret1 = 0; 1; 3; 5; 13, tl = 0; 1; . . . ; 88 for l = 2; 3; 4 and
k = 0; 1; . . . ; 22.
Applying the orthogonality condition (1) we obtain the elements of the generator matrix ofM2. UsingM1andM2, we construct the gen-erator matrix ofE(C)3and withgen(F(C)) from (3) we obtain the
generator matrix of the codeC.
A computer search shows that in each case, the code C has a vector of weight less than20. Hence, the binary [96; 48; 20] self-dual doubly-even code cannot have an automorphism of order 23. The proposition is proved.
Proposition 2: A self-dual doubly-even[96; 48; 20] code C cannot
have an automorphism of order11.
Proof: SupposeC has an automorphism of type 11-(8; 8).
Then,(F(C)) is a binary [16; 8] self-dual code. Denote by b20the number of weight-20 vectors of the code C and by c20the number of weight–20 vectors of F(C).
The permutation splits the vectors of the code C in orbits of length
1 or 11. Any vector of F(C) is in an orbit of length 1. A vector of
E(C) is in an orbit of length 1 if and only if it is the all-zero vector.
Therefore,b20 c20(mod 11) or c20 7(mod 11), because
b20= 3217056 7(mod 11):
Since20 = 0 2 11 + 20 and 20 = 1 2 11 + 9 it follows that the number of fixed points must be20 or 9. Then, F(C) does not contain a
weight–20 vector, because the automorphism has eight fixed points. Thus,c20 = 0. This contradicts to c20 7(mod 11). Hence, the codeC cannot have an automorphism of order 11.
Proposition 3: A self-dual doubly-even[96; 48; 20] code C cannot
have an automorphism of order7.
Proof: According to [11], there are two possibilities for an
auto-morphism of order7.
First, supposeC has an automorphism of type 7-(13; 5). Then
(F(C)) is a binary [18; 9] self-dual code. All such codes were
clas-sified by Pless [6]. If the[2; 1] code C2 withgen(C2) = (11) is a
direct summand of(F(C)) then the subcode F(C) will contain a
vector with weight in the setf2; 8; 14g, but the minimum distance of the codeC is 20. According to [6], (F(C)) is either H18orI18with generator matrices gen(H18) = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
TABLE I and gen(I18) = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :
Next we recall the notions of a duo and a cluster.
A duo is any set of two coordinate positions of a code. A cluster is a set of disjoint duos such that the union of any two duos is support of a weight–4 vector of the code.
Any weight–4 vector of (F(C)) generates a vector of F(C) with
weight in the setf4; 10; 16; 22; 28g. Only the last value is possible. Suppose(F(C)) = H18. Clearly, the sets
M1= ff1; 2g; f3; 4g; f5; 6gg
M2= ff7; 8g; f9; 10g; f11; 12gg
and
M3= ff13; 14g; f15; 16g; f17; 18gg
are clusters forH18or every 18 positions cannot be fixed. There are five fixed points by and this excludes H18.
Consider(F(C)) = I18. The sets
M4= ff1; 2g; f3; 4g; f5; 6g; f7; 8g; f9; 10gg
and
M5= ff12; 13g; f14; 15g; f16; 17gg
are clusters and hence only two positions (11th and 18th) can be fixed. Therefore, a binary[96; 48; 20] self-dual doubly-even code cannot have an automorphism of type7-(13; 5).
Assume now thatC has an automorphism of type 7-(12; 12). Then,
(F(C)) is a binary [24; 12] self-dual code.
It is proved in [8] that a generator matrix of a self-dual[n; n=2] code can be chosen in the form
A 0
0 B
D E
(7)
whereA, B, D, and E are matrices of types k12n1,k22n2,k32n1, andk32n2, respectively. MatricesD and E do not have zero rows and alsok1+ k2+ k3= n=2 and k2= k + k10 n1.
Consider a generator matrix of(F(C)) in the form (7), where A,
B, D, and E are matrices of types k1 2 12, k22 12, k32 12, and
k32 12. Then k1+ k2+ k3 = 12 and k2 = 12 + k10 12. Hence
k1 = k2.
Obviously, the matrix B must generate a [12; k2; d1] code with
d1 20. It follows that k2 = 0 and also k1 = 0.
A weight–4 vector of (F(C)) generates only weight–28 vector of
F(C) and then k1 > 0. Therefore, (F(C)) cannot have a weight–4
vector. Hence,(F(C)) is either [24; 12; 6] or [24; 12; 8] self-dual
code. Up to equivalence, the extended Golay[24; 12; 8] code (G24)
and the[24; 12; 6] code Z24are unique (see [7]). A computer check shows that if(F(C)) is either G24 orZ24 then the codeF(C)
contains a vector of weight less than20 or a vector of a weight not divisible by4. Hence, the code C cannot have an automorphism of type7-(12; 12) either.
III. NEW[96; 48; 16] SELF-DUALDOUBLY-EVENCODES
LetD be a [96; 48; 16] self-dual doubly-even code. The weight enu-merator for the codeD is given in [2] and it is
W (y) = 1 + (028086 + )y16+ (3666432 0 16 )y20
+(366474560 + 120 )y24+ 1 1 1 : (8)
The first such code was found by Feit [3] and it has a weight enumer-ator (8) for = 37 722. Recently, Dougherty, Gulliver, and Harada [2] found four new inequivalent[96; 48; 16] self-dual doubly-even codes for = f37 584; 37 500; 37 524; and 37 598g.
Here we consider 25 new codes via an automorphism of order23. Consider a[96; 48; 16] code D possessing an automorphism of order
23 with four cycles and four fixed points. We choose the generator
matrix ofF(D) in the form
gen(F(D)) =
a 1 0 0 0
a 0 1 0 0
a 0 0 1 0
a 0 0 0 1
wherea is the all-one vector of length 23 and where nonindicated en-tries are equal to zero (asgen(F(C)) from (3)).
We have'(E(D)3) = M18 M2anddim M1= 3 or 2. Hence,
the generator matrix ofM1can have the forms (4)–(6).
Applying the orthogonality condition (1), we obtain the corre-sponding generator matrices forM2. In this way, we obtain a generator matrix of the codeD in the form
a 1 0 0 0 a 0 1 0 0 a 0 0 1 0 a 0 0 0 1 u w1 0 0 0 0 u w2 0 0 0 0 u w3 0 0 0 0 w4 w5 w6 v 0 0 0 0 and a 1 0 0 0 a 0 1 0 0 a 0 0 1 0 a 0 0 0 1 u r1 r2 0 0 0 0 u r3 r4 0 0 0 0 r5 r6 v 0 0 0 0 r7 r8 v 0 0 0 0
wherea is the all-one vector of length 23, nonindicated entries are equal to zero, and the cellsu, v, wifori = 1; . . . ; 6, and rjforj = 1; . . . ; 8 are11 2 23 circulant matrices.
TABLE II
By computer check fort1= 0, t2= 1; 2; 3; and t3= 0; 1; . . . ; 88 in (4) we obtain 22 inequivalent binary [96; 48; 16] self-dual doubly-even codes. For some values oft1,t2,t3,t4, andk in (6) we obtain three more codes.
We express the first rows for the cellsu, v, wifori = 1; . . . ; 6, andrjforj = 1; . . . ; 8 by 0 = 0000,1 = 0001,. . ., 9 = 1001, A =
1010, B = 1100, . . . , F = 1111 in hexadecimal form. The vectors
of the first rows are placed immediately one after another, omitting the last zero.
The cellsu and v begin respectively with 85335E and F ACCA0. The values of in the weight enumerator (8) and in the first rows for cellswifori=1; . . . ; 6 and for cells rjforj =1; . . . ; 8 in each of the obtained codes are given in Tables I and II.
ACKNOWLEDGMENT
The author wishes to thank the referees for their detailed remarks and Prof. S. Dodunekov for helpful discussions.
REFERENCES
[1] S. Bouyuklieva, “A method for constructing self-dual codes with an automorphism of order 2,” IEEE Trans. Inform. Theory, vol. 46, pp. 496–504, Mar. 2000.
[2] S. T. Dougherty, T. A. Gulliver, and M. Harada, “Extremal binary self-dual codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 2036–2047, Nov. 1997.
[3] W. Feit, “A self-dual even (96,48,16) code,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 136–138, Jan. 1974.
[4] W. C. Huffman, “Automorphisms of codes with application to extremal doubly-even codes of length 48,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 511–521, May 1982.
[5] C. L. Mallows and N. J. A. Sloane, “An upper bound for self-dual codes,” Inform. Contr., vol. 22, pp. 188–200, 1973.
[6] V. Pless, “A classification of self-orthogonal codes over GF(2),” Discr. Math., vol. 3, pp. 209–246, 1972.
[7] , “On the classification and enumeration of self-dual codes,” J. Comb. Theory, vol. 18, pp. 313–335, 1975.
[8] V. Pless, N. J. 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, May 1980.
[9] V. Y. Yorgov, “Binary self-dual codes with automorphisms of odd order,” Probl. Pered. Inform., vol. 19, pp. 11–24, 1983. Russian.
[10] , “A method for constructing inequivalent self-dual codes with ap-plications to length 56,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 77–82, Jan. 1987.
[11] V. Y. Yorgov, A. Shterev, and N. Ziapkov, “A [96, 48, 20] doubly-even self-dual code does not have automorphism of order 47 and 31,” in Int. Workshop ACCT, Leningrad, U.S.S.R., 1990, pp. 191–194.
