On the [28, 7, 12] binary self-complementary codes and their residuals

Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA0=A28=1,A12=A16=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.

[1]  Tom Verhoeff,et al.  An updated table of minimum-distance bounds for binary linear codes , 1987, IEEE Trans. Inf. Theory.

[2]  R. A. Rankin,et al.  On the minimal points of positive definite quadratic forms , 1956 .

[3]  William M. Kantor,et al.  Exponential numbers of two-weight codes, difference sets and symmetric designs , 1983, Discret. Math..

[4]  Vera Pless,et al.  A classification of self-orthogonal codes over GF(2) , 1972, Discret. Math..

[5]  Andries E. Brouwer,et al.  Some unitals on 28 points and their embeddings in projective planes of order 9 , 1981 .

[6]  M. Shrikhande,et al.  Quasi-Symmetric Designs , 1991 .

[7]  Jennifer D. Key,et al.  Arcs and Ovals in the Hermitian and Ree Unitals , 1989, Eur. J. Comb..

[8]  R. Rankin.  The Closest Packing of Spherical Caps in n Dimensions , 1955, Proceedings of the Glasgow Mathematical Association.

[9]  Vladimir D. Tonchev,et al.  On Symmetric and Quasi-Symmetric Designs with the Symmetric Difference Property and Their Codes , 1992, J. Comb. Theory A.

[10]  C. A. V. T. Henk.  The smallest length of binary 7-dimensional linear codes with prescribed minimum distance , 1981 .

[11]  Vladimir D. Tonchev,et al.  Designs with the Symmetric Difference Property on 64 Points and Their Groups , 1994, J. Comb. Theory A.

[12]  Peter Blundell Gibbons,et al.  Computing techniques for the construction and analysis of block designs. , 1976 .

[13]  Vladimir D. Tonchev,et al.  Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound , 1991, Des. Codes Cryptogr..

[14]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[15]  L. D. Grey.  Some bounds for error-correcting codes , 1962, IRE Trans. Inf. Theory.

[16]  V. Tonchev.  Quasi-symmetric designs, codes, quadrics, and hyperplane sections , 1993 .

[17]  William M. Kantor,et al.  Symplectic groups, symmetric designs, and line ovals , 1975 .

[18]  Hanfried Lenz,et al.  Design theory , 1985 .