Research On Some Problems Of Linear Codes Over Finite Ring

In the 1990s, the theory of error-correcting codes over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes over Z₄under the Gray map. Since then, codes over finite rings have been given more attention. In this paper, we study the MacWilliams identities for different weight enumerators of the linear codes and their dual codes over F₂ [u ]/ (u^k+1 )and(1 + u^k)constacyclic codes, and investigate some properties of self-dual codes and cyclic codes over F₂ + vF₂. We also study the one and two-Lee weight codes over Z₄. The details are given as follows:(1) Using the weight distribution relations between binary linear codes and their dual codes and new Gray map, The MacWilliams identities for different weight enumerators of the linear codes and their dual codes over F₂ + uF₂ + + u^kF₂ were given.(2) A new mapφ^k from F₂ + uF₂ + + u kF₂ to F₂ + uF₂ is defined, and proved that the Gray image of a (1 + u^k)constacyclic code over the ring is a binary quasi-cyclic code of index 2^k-1 and length 2k n.(3) A necessary and sufficient condition for the existence of Euclidean self-dual codes and Type II codes over the ring F₂ + vF₂ are obtained respectively, a method of constructing self-dual codes over the ring is given.(4) We prove that cyclic code over F₂ + vF₂ is a principal ideal of Rn = R[ x ]/( xn? 1), the generator polynomial of cyclic codes over the ring is obtained. Cyclic codes with odd length over F₂ + vF₂ have a unique idempotent polynomial is determined.(5) Some properties of one and two Lee-weight codes over Z₄are obtained, and a necessary and sufficient condition for the existence of one Lee-weight codes and two Lee-weight projective codes over Z₄ is given respectively.