Research On The Covering Radius Of Codes Over Finite Non Chain Rings

Error-correcting coding theory is the theoretical bases of inofmration safety. At present, error-correcting codeing theory over finite fields has been not only developed perfecting but also applied widely to the productive practice. With the successive development of production technique and the successive deepgoing researches on theory.Researches on error-correcting coding theory over finite rings have not only important theoretical significance but also important practical value.In1994, it was shown that certain good nonlinear binary codes can be constructed from cyclic codes over Z4via the Gray map by Hammons, et al. Subsequently, many improtant results over the ring Z4were obtained. The covering radius of a code is a fundamental geometric parameter of a code,characterizing its maximal error correcting capability. The topic has applications to problems of data compression, testing, and write-once memories, which is a research hotspot in recent years. The covering radius of codes over some finite chain rings such as Z4and F2+uF2has been studied. On the contrary there are a few studies on the covering radius of codes over finite non chain rings. In this paper, we mainly study the covering radius of codes over the ring F2+vF2and Z3[v]/<v3-v>. The details are given as follows:1. We study the covering radius of codes over the ring F2+vF2for the Lee distance. We give a distance preserving Gray map for the Lee distance and study the structure of linear codes over F2+vF2. The covering radius of codes over this ring is defined. It is proved that the covering radius of a linear code and its dual codes over this ring is equal to the sum of the covering radius of two binary codes. Several upper and lower bounds on the covering radius are given.2. We introduce a Gray map from Z3[v]/<v3-v> to Z33and give a relation between codes over the ring and ternary codes. The Gray weight of elements of Z3[v]/<v3-v> is defined. Generating sets of Gray image of linear codes over this ring are given. We define the concept of the subcode and study the relation between the covering radius of a linear code and subcodes, the covering radius of the dual codes over this ring is also discussed.