The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over F5 for lengths 22, 26, 28, 32 − 40. In particular, we prove that there is no [22, 11, 9] self-dual code over F5, whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over F5 are un...

It is shown that some well-known and some new cyclic codes with orthogonal parity-check equations can be constructed in the finite-field transform domain. It is also shown that, for some binary linear cyclic codes, the performance of the iterative decoder can be improved by substituting some of the dual code codewords in the parity-check matrix with other dual code codewords formed from linear ...

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C1 and C2, are equivalent if and only if there is a permutation of the coordinates of C1 that takes C1 into C2. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself. Th...

An (n, k) binary self-orthogonal code is an (n, k) binary linear code C that is contained in its orthogonal complement C⊥. A self-orthogonal code C is self-dual if C = C⊥. Two codes, C1 and C2, are equivalent if and only if there exists a coordinate permutation of C1 that takes C1 into C2. The automorphism group of a code C is the set of all coordinate permutations of C that takes C into itself...

This report is a survey of self-dual binary codes. We present the fundamental MacWilliams identity and Gleason’s theorem on self-dual binary codes. We also examine the upper bound of minimum weights of self-dual binary codes using the extremal weight enumerator formula. We describe the shadow code of a self-dual code and the restrictions of the weight enumerator of the shadow code. Then using t...

One of the most remarkable theorems in coding theory is Gleason’s 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes, always on a case-by-case basis. In this talk I will state the theorem and then describe the far-reaching generalization t...

The Kerdock and extended Preparata codes are something of an enigma in coding theory since they are both Hamming-distance invariant and have weight enumerators that are MacWilliams duals just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparata-like code PL from the Kerdock code K, why the existence of a distance-invariant code with weight d...

There is a one-to-one correspondence between l-quasi-cyclic codes over a finite field Fq and linear codes over a ring R = Fq[Y ]/(Y m − 1). Using this correspondence, we prove that every l-quasi-cyclic self-dual code of length ml over a finite field Fq can be obtained by the building-up construction, provided that char (Fq) = 2 or q ≡ 1 (mod 4), m is a prime p, and q is a primitive element of F...