Research On Linear Codes And Self-dual Codes Over Finite Rings

Theoretics foundation of coding theory is based on mathematics . However, this theory is applied in many situations which have as a common feature that information coming from some source is transmitted over a noisy communication channel to a receive . With the development of information technology, coding theory makes progress rapidly. In the 1980s, 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 Z 4 under the Gray map. Since then, codes over finite rings have been given more attention.In the present paper , anthor studys some linear error-correcting codes. Author does some work about the kind of codes over rings. The content is as follow :1 We study on condition of existence of self-dual code over F2 + uF2 + + u kF2. If k is odd, we prove that self-dual code must exist; if k is even, we give a necessary and sufficient condition for the existence of self-dual code over ring F2 + uF2 + + u kF2.2 We popularize the concept of the residue code and torsion code of cyclic code from over ring F2 + uF2 to ring F₂ + uF₂ + + u kF₂. We give the definition of generator matrix of code and high-end torsion code, then prove some properties of self-dual code's type .3 We give the definition of a mappingφfrom ring ( ) to ring When the code C is the length n of self-dual code over ring , we proveφ(C ) is the length n of self-orthogonal over ring F₂ + uF₂ and get that the necessary and sufficient condition ofφ(C ) which is the length n of self-dual code over ring F2 + uF2.4 We give the definition of Gray mapΦover ring F₂ + uF₂ + + u^kF₂ . When the code C is the length n of self-dual code over ring F₂ + uF₂ + + u^kF₂ and k≥3, we prove images of the Gray map are self-orthogonal codes.