Error-block codes and poset metrics

Abstract

Let P = ({1, 2, n},≤) be a poset, let V1, V2, Vn be a family of finite-dimensional spaces over a finite field Fq and let V = V1 ⊕ V2 ⊕ . . . ⊕ Vn. In this paper we endow V with a poset metric such that the P-weight is con-stant on the non-null vectors of a component Vi, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code [8; 4; 4] to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code [24; 12; 8] into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces. ©2008 AIMS-SDU.