A. Extension theorems for linear codes over finite rings.pdf

Extension Theorems for Linear Codes overFinite RingsJay A. Wood?Department of Mathematics, Computer Science StatisticsPurdue University CalumetHammond, Indiana 46323{2094 USAwood@Abstract. Various forms of the extension problem are discussed forlinear codes dened over nite rings. The extension theorem for sym-metrized weight compositions over nite Frobenius rings is proved. As aconsequence, an extension theorem for weight functions over certain -nite commutative rings is also proved. The proofs make use of the linearindependence of characters as well as the linear independence of charac-ters averaged over the orbits of a group action.1 IntroductionWitt [14] and Arf [1] were among the rst mathematicians to prove extensiontheorems in algebra. If V is a nite dimensional vector space over a eld and ifV is equipped with a non-degenerate quadratic form Q, then for any subspaceW  V , Arf proved that every injective linear transformation f :W ! V whichpreserves Q extends to a linear automorphism of V which preserves Q.MacWilliams was the rst mathematician to prove an extension theorem incoding theory [9], [10]. If V is a nite dimensional vector space over a nite eld,then a choice of basis determines the Hamming weight on V . For any subspaceW  V and any (injective) linear transformation f : W ! V which preservesHamming weight, MacWilliams proved that f extends to a linear automorphismof V which preserves Hamming weight; i.e., the extension of f is a monomialtransformation. This theorem became the cornerstone of the idea of equivalenceof codes.There have been other proofs of this extension theorem of MacWilliams, forexample [3], [6], [13]. Motivated by the increased interest in linear codes denedover nite rings sparked by the famous ZZ=4-paper [8], the author proved theextension theorem for Hamming weight over nite Frobenius rings [15].All the results mentioned above deal with the Hamming weight. Other weightfunctions such as the Lee weight are also importan