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Error-block codes and poset metrics. (English)
Adv. Math. Commun. 2, No. 1, 95-111 (2008).
Summary: Let $P = (\{1, 2, \ldots, n\},\leq )$ be a poset, let $V_1 , V_2 , \ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\Bbb F_q$ and let $$V = V_1\oplus V_2 \oplus \ldots \oplus V_n .$$ In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, 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.