Group convolutional codes (Q939250)

As a generalization of both cyclic convolutional code and group block code, group convolutional code is defined in this article. Let \(F\) be a finite field and \(G\) a finite group of order \(n\) such that the characteristic of \(F\) does not divide \(n\). The group algebra \(A=F[G]\) is isomorphic to \(F^n\) via \((v_1,\dots,v_n)\mapsto \sum_i v_ig_i\) where \(g_1,\dots,g_n\) is a fixed order of the elements of \(G\). Since \(F^n[z]\) and \(F[z]^n\) are naturally isomorphic, \(F[z]\)-submodules of \(F[z]^n\) and \(F[z]\)-submodules of \(A[z]\) can be identified. Let \(R=A[z;\sigma]\) be the skew polynomial ring, where \(\sigma\) is an \(F\)-automorphism of \(A\). An \(F[z]\)-submodule \(C\) of \(F[z]^n\) is said to be a \((G,\sigma)\)-convolutional code if \(C\), considered as a left ideal in \(R\), is a direct summand. A \((G,\sigma)\)-convolutional code is said to be minimal if it is indecomposable as left \(A[z;\sigma]\)-module. The main part of the article is devoted to the determination of the minimal \(S_3\)-convolutional code over the field \(F_5\) with five elements using Jategaonkar's theorems [\textit{A. V. Jategaonkar}, J. Algebra 19, 315--328 (1971; Zbl 0223.16005)] which describe skew polynomial rings in terms of matrices. The conclusions are (1) all the minimal \(S_3\)-convolutional codes over \(F_5\) obtained have the parameters \((6,1,t)\) or \((6,2,2t)\) just like that of minimal \(Z_6\)-convolutional codes, and (2) \(S_3\)-convolutional codes are not necessary block codes when \(\sigma=id\), while \(\sigma\)-cyclic convolutional codes are always block codes when \(\sigma=id\).