A lower bound on the weight hierarchies of product codes.
The concepts of generalized Hamming weight and weight hierarchy of an error correcting code were first introduced by \textit{V. K. Wei} [Generalized Hamming weights for linear codes, IEEE Trans. Inf. Theory 37, 1412--1418 (1991; Zbl 0735.94008)], motivated by the cryptographic problem of the wire-tap channel of type II. The weight hierarchy of an \([n,k,d]\) linear code \({\mathcal C}\) is the sequence \(d_1=d <d_2 <\cdots <d_k=n\), where \(d_r\), the \(r\)th generalized Hamming weight, is the minimum support weight of the \(r\)-dimensional subcodes of \({\mathcal C}\). In that paper Wei proposed, among other open problems, the characterization of the weight hierarchy of a product code \({\mathcal C}={\mathcal C}_1\otimes {\mathcal C}_2\) in terms of the hierarchies of the factor codes \({\mathcal C}_i\). \textit{V. K. Wei} and \textit{K. C. Yang} [On the generalized Hamming weight of product codes, IEEE Trans. Inf. Theory 39, 1709--1713 (1993; Zbl 0801.94016)] establish a conjecture for such a hierarchy providing that the factor codes satisfy a certain \` \` chain condition''. Many authors, as those mentioned in the references of this paper, have contributed to solving some particular instances of the conjecture, which was finally demonstrated in an independent way, and using different techniques, by both authors of this paper [\textit{H. G. Schaathun}, The weight hierarchy of product codes, IEEE Trans. Inf. Theory 46, 2648--2651 (2000; Zbl 1001.94053); \textit{C. Martinez-Perez} and \textit{W. Willems}, On the weight hierarchy of product codes, preprint (2001)]. The present authors demonstrate, using the techniques from the paper of Schaathun, a generalization of the conjecture. They consider the product of \(t\) codes \({\mathcal C}={\mathcal C}_1\otimes \cdots \otimes{\mathcal C}_t\) and they give the following result (\(d_r^{*}\) was defined in the paper of Wei and Yang): \(d_r({\mathcal C})\geq d_r^{*}({\mathcal C}),\, r=1, \dots, k\). Moreover equality holds if all the components \({\mathcal C}_i\) are chained. Finally the authors suggest some new open problems.
scientific article; zbMATH DE number 1925286
scientific article