A commutative algebra approach to linear codes
The authors generalize the notion of a cyclic code which allows the extension of computational algebra techniques to larger classes of codes. A linear code over the field GF\((q)\) is called an \(n\)-th \textit{root code} if it has, as parity-check matrix, a matrix \(H=(g_i(\ell_j))_{{i=1,\dots,r\atop j=1,\dots,N}}\) where \(\ell_1,\dots,\ell_N\) are different and \(n\)-th roots of unity (or zero) and lie in an extension field \(F=\)GF\((q^m)\) and where \(g_1,\dots,g_r\) are polynomials over \(F\) such that for all \(j\in\{ 1,\dots,N\}\) there is at least one \(i\in\{ 1,\dots,r\}\) with \(g_i(\ell_j)\neq 0\). It is shown that any linear code is an \(n\)-th root code (and vice versa). Given a representation of a code \(\mathcal{C}\) as \(n\)-th root code, distance and weight distribution of \(\mathcal{C}\) can be computed via Gröbner basis techniques. The decoding techniques of \textit{E. Orsini} and \textit{M. Sala} [J. Pure Appl. Algebra 200, No. 1--2, 191--226 (2005; Zbl 1159.94397)] via the general error locator polynomial can be extended to a large class of \(n\)-root codes (containing the ``proper maximal zerofree'' codes having \(\ell_i\neq 0\) for \(i=1,\dots,N\)) . An explicit example is given to show that, depending on the representation as \(n\)-th root code, its error locator polynomial can be small or long.
scientific article; zbMATH DE number 5565631
scientific article