Two Classes Of Three-weight Codes And Their Relationship With Association Scheme

Since coding theory emerged at the end of the1940s, computing the weight distributionof codes has become the main part of men’s study. Association scheme appeared and grew upquickly in1970s, and was used widely in coding theory, design theory, the graph theory and soon. This paper introduces two classes of linear codes and describes the relation between twoclasses of linear codes and the association scheme. Let p is a prime, q=ps, r=qm, wheres, m are positive integers. First, in this paper we construct a class of three weight code C froma class of three weight cyclic codes on the finite field Fq. The restriction to this kind of codes Cof Hamming scheme need not be an association scheme(That is, this kind of codes need not de-fine association scheme). Second, we compute the weight distribution of a kind of three cycliccodes on the finite field Fq, where the code length is n. The parity polynomial of this kind ofthree weight codes is h(x)=x~3a(a∈Fq), where h(x)|x~n1. We calculate the weightdistributions of the coset of its dual codes x+C~⊥(x∈F_q~n), and get four classes of differentweight distributions, which proves that the restriction to this kind of codes of Hamming schemeis an association scheme(That is, this kind of codes can define association scheme). Finally, thiskind of cyclic codes is extended, and we compute the weight distribution of the codes havingparity polynomials h(x)=x~t a(a∈F_q, t≥2), and the restriction to this extended code ofHamming scheme is itself an association scheme.