On [formula omitted]-cyclic codes and their applications in constructing optimal codes

Abstract

Let R=F2+uF2(u2=0) and S=F2+uF2+u2F2(u3=0) be two finite commutative chain rings. This paper studies F2RS-cyclic codes, which are described as S[x]-submodules of the S[x]-module F2[x]∕〈xr−1〉×R[x]∕〈xs−1〉×S[x]∕〈xt−1〉. We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free F2RS-cyclic codes and separable codes are discussed, and the structural properties of dual codes of free F2RS-cyclic codes are investigated. Moreover, we determine the relationship between the generator polynomials of free F2RS-cyclic codes and their duals. As applications, we provide several examples of optimal and near-optimal binary codes which are obtained from the Gray images of F2RS-cyclic codes.