HIGHER WEIGHTS AND GENERALIZED MDS CODES

Abstract

We study codes meeting a generalized version of the Single- ton bound for higher weights. We show that some of the higher weight enumerators of these codes are uniquely determined. We give the higher weight enumerators for MDS codes, the Simplex codes, the Hamming codes, the flrst order Reed-Muller codes and their dual codes. For the putative (72;36;16) code we flnd the i-th higher weight enumerators for i = 12 to 36. Additionally, we give a version of the generalized Singleton bound for non-linear codes. Maximum Distance Separable (MDS) codes are an important class of codes. They are signiflcant since they are optimal codes for their length and dimension but also because of their relation to various combinatorial structures, see (11) for a complete description. In this paper, we study a generalization of MDS codes, namely those codes meeting a generalized Singleton bound for higher weights. We shall deflne higher weights and then describe some basic results of generalized MDS codes which we will generalize. We also give the higher weight enumerators for the Simplex codes, the Hamming codes, the flrst order Reed-Muller codes and their dual codes. We give a version of the generalized Singleton bound for non-linear codes. We use these results to flnd a signiflcant number of the higher weight enumerators for the putative (72;36;16) Type II code. We flnd the i-th higher weight enumerators for i = 12 to 34 as well using the techniques of the paper. We also use the techniques of the paper to determine some previously unknown higher weight enumerators of the codes associated with the projective plane of order 5. We begin with some basic deflnitions from coding theory. For any undeflned terms from coding theory see (11) or (18). A code C of length n overFq is a subset ofF n, and if it is a vector space, then we say it is a linear code. We attach the standard inner product to the ambient space F n, that is (v;w) = P viwi; and deflne C ? = fv j (v;w) = 0 for all w 2 Cg: If C = C ? , then C is said to be a self-dual code. A Type II