# American Institute of Mathematical Sciences

February  2008, 2(1): 95-111. doi: 10.3934/amc.2008.2.95

## Error-block codes and poset metrics

 1 Centro Politécnico, UFPR, Caixa Postal 019081, Jd. das Américas, CEP 81531-990, Curitiba, PR, Brazil 2 Centro de Engenharias e Ciências Exatas, UNIOESTE, Av. Tarquínio Joslin dos Santos, 1300, CEP 85870-650, Foz do Igua¸cu, PR, Brazil 3 Imecc - Unicamp, CP 6065, CEP 13083-970, Campinas, SP, Brazil

Received  September 2007 Revised  January 2008 Published  January 2008

Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ 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.
Citation: Marcelo Muniz S. Alves, Luciano Panek, Marcelo Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2008, 2 (1) : 95-111. doi: 10.3934/amc.2008.2.95
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