The Study Of Depth Of Codes Over Ring

In recent years,many researchers on coding theory are interesting instudying coding theory over finite rings ,especially in studying Z₄ -code.In 1997,T.Etzion introduced depth of codes over fields and some properties in [14]. He also gave the recursive algorithms for computing thedepth of codewords over F₂ .In [15] ,C.J.Mitchell studied the depth spectrum of cyclic codes over F₂. In 2000,Luo gave the recursive algorithms forcomputing the depth of codewords over field F_q .In this dissertation, we studythe depth distribution of linear codes over rings and algorithms of codewords over rings, and the main results are as follows:1. Theory of depth distribution of codes over a finite ring Z₄ is established. We defin.e the depth of a codeword and the depth distribution of codes over a finite ring Z₄ ,and give a number of properties on the depth of codewords and the depth distribution of codes on ring Z₄ ,and it si proved that the depth spectrum of linear codes of type 4^k+12^k₂ has at least k₁+k₂ nonzero values ,and the depth spectrum of linear cyclic codes of type 4^k is {n,n-1,...,n-k + 1},and studies the depth spectrums of a special type of Z₄ - negacyclic codes. By using combinatory mathematics ,theory of graph and number theory, two recursive algorithms for computing the depth of a codeword on ring Z₄ are presented.2. The depth of a codeword over ring Z _p^k is definited, and a number ofproperties of depth distribution of linear codes over ring Z _p^k are studied. Arecursive algorithm for computing the depth of codewords over ring Z _p^k isalso given.