This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal I + (C) to an arbitrary linear code. The binomials involved in the reduced Gröbner basis of such an ideal relative to a degreecompatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of I + (C) provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.2010 Mathematics Subject Classification. Primary: 94B05, 13P25; Secondary: 13P10.

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Hadamard full propelinear codes (HFP-codes) are introduced and their equivalence with Hadamard groups is proven (on the other hand, it is already known the equivalence of Hadamard groups with relative (4n, 2, 4n, 2n)-difference sets in a group and also with cocyclic Hadamard matrices). We compute the available values for the rank and dimension of the kernel of HFP-codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP-code of length 4n, dimension of the kernel k = 2, and maximum rank r = 2n, we obtain an HFP-code of double length 8n, dimension of the kernel k = 2, and maximum rank r = 4n.

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