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Error-block codes and poset metrics

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  • 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.
    Mathematics Subject Classification: Primary: 94B05, 06A06; Secondary: 20B30.

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