On Toeplitz codes of index t and isometry codes

Abstract

In this paper, we introduce Toeplitz codes of index t, which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1t over Fq, so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [32,16,8], [33,11,11] and [34,17,8] codes with improved parameters directly.