Shadow Codes over Z4

The notion of a shadow of a self-dual binary code is generalized to self-dual codes over Z4. A Gleason formula for the symmetrized weight enumerator of the shadow of a Type I code is derived. Congruence properties of the weights follow; this yields constructions of self-dual codes of larger lengths. Weight enumerators and the highest minimum Lee, Hamming, and Euclidean weights of Type I codes of length up to 24 are studied.

[1]  P. Gaborit,et al.  All Self-Dual Z4 Codes of Length 15 or Less Are Known , 1997, IEEE Trans. Inf. Theory.

[2]  N. J. A. Sloane,et al.  A new upper bound on the minimal distance of self-dual codes , 1990, IEEE Trans. Inf. Theory.

[3]  Han-Ping Tsai.  Existence of some extremal self-dual codes , 1992, IEEE Trans. Inf. Theory.

[4]  Michael Klemm.  Über die Identität von MacWilliams für die Gewichtsfunktion von Codes , 1987 .

[5]  Christine Bachoc,et al.  Type II codes over Z4 , 1997, IEEE Trans. Inf. Theory.

[6]  E. M. Rains,et al.  Self-Dual Codes , 2002, math/0208001.

[7]  Masaaki Harada,et al.  Type II Codes, Even Unimodular Lattices, and Invariant Rings , 1999, IEEE Trans. Inf. Theory.

[8]  Masaaki Harada,et al.  Certain Self-Dual Codes over Z4 and the Odd Leech Lattice , 1997, AAECC.

[9]  Eric M. Rains.  Optimal self-dual codes over Z4 , 1999, Discret. Math..

[10]  N. Sloane,et al.  On the enumeration of lattices of determinant one , 1982 .

[11]  A. Robert Calderbank,et al.  Quaternary quadratic residue codes and unimodular lattices , 1995, IEEE Trans. Inf. Theory.

[12]  N. J. A. Sloane,et al.  Self-Dual Codes over the Integers Modulo 4 , 1993, J. Comb. Theory, Ser. A.

[13]  Steven T. Dougherty.  Shadow codes and weight enumerators , 1995, IEEE Trans. Inf. Theory.

[14]  N. J. A. Sloane,et al.  A linear construction for certain Kerdock and Preparata codes , 1993, ArXiv.

[15]  Richard A. Brualdi,et al.  Weight enumerators of self-dual codes , 1991, IEEE Trans. Inf. Theory.

[16]  V. Pless,et al.  Cyclic Self-DualZ4-Codes , 1997 .

[17]  N. J. A. Sloane,et al.  An Upper Bound for Self-Dual Codes , 1973, Inf. Control..