The weight enumerator of linear codes over \(\mathrm{GF}(q^m)\) having generator matrix over \(\mathrm{GF}(q)\)
Let \(C\) be an \([n,k]\)-code over \(\mathrm{GF}(q)\) with generator matrix \(G\). Let \(C^{(m)}\) denote the code over \(\mathrm{GF}(q^m)\) generated by the same matrix. \textit{T. Helleseth}, \textit{T. Kløve} and \textit{J. Mykkeltveit} [Discrete Math. 18, 179--211 (1977; Zbl 0357.94007)] gave an expression for the (Hamming) weight enumerator of \(C^{(m)}\) involving the number of \((k-j)\)-dimensional subspaces of \(\mathrm{GF}F(q)^k\) that contain exactly \(n-i\) columns of \(G\). Here, an alternative expression for this weight enumerator is given by using the \(\lambda\)-ply weight enumerator introduced by the author in a previous paper [Hokkaido Math. J. 25, 651--656 (1996; Zbl 0868.05006)]. As an application, the author determines the support weight enumerator (giving the number of \(r\)-dimensional subcodes of \(C\) with support weight \(i\)) for MDS codes.
scientific article; zbMATH DE number 1275106