A lower bound on the greedy weights of product codes (Q1431622)

The concepts of generalized Hamming weights and weight hierarchy of a linear error correcting code were introduced by \textit{V. K. Wei} [IEEE Trans. Inf. Theory 37, 1412--1418 (1991; Zbl 0735.94008)]. 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 rth generalized Hamming weight, is the minimum support weight of the r-dimensional subcodes of \({\mathcal C}\). In that paper Wei proposed, as an open problem, the characterization of the weight hierarchy of a product code (the tensor product of two codes) as a function of the hierarchies of the factor codes. \textit{V. K. Wei} and \textit{K. Yang} [ibid. 39, 1709--1713 (1993; Zbl 0801.94016)] establish a conjecture for such a hierarchy providing that the factor codes satisfy a certain \` \` chain condition''. Several authors have contributed to the study of this conjecture, which was finally demonstrated in an independent way by the author of this paper alone [\textit{H. G. Schaathun}, ibid. 46, 2648--2651 (2000; Zbl 1001.94053)] and in collaboration with \textit{C. Martinez-Perez} and \textit{W. Willems} [Electron. Notes Discrete Math. 6 (2001; Zbl 0987.94516)]. A lower bound for the Hamming weight \(d_i\) of a product code is obtained and equality is reached if the factor codes satisfy the chain condition. The present paper is concerned with a refinement of the generalized Hamming weights \(d_i\), the greedy weights \(e_i\)\, and also with the related concept of top-down greedy weights. The greedy weights \(e_i\)\, of a code coincide with \(d_i\) if and only if the code satisfies the chain condition. In the paper the author proves a lower bound for the greedy weight \(e_i\) of a product code (theorem 2 of the paper) similar to the bound he obtained for the generalized Hamming weight \(d_i\) (recalled in theorem 1).