Research On Constacyclic Codes Over Some Classes Of Finite Non-chain Rings

Cyclic codes not only have good algebraic structures,but also can easily be encoded and decoded.Therefore,whether on finite fields or finite rings,they are widely researched by communication and coding scholars.As the generalization of cyclic codes,constacyclic codes naturally have important research values.Because of their special algebraic structures,constacyclic codes over finite fields and finite chain rings have been widely studied.However,there has been a limited study on constacyclic codes over finite non-chain rings.In this thesis,the structure and some related problems of constacyclic codes over finite non-chain rings F2+uF2+vF2are studied,and by Gray map,We study the relations between codes over this ring and codes over the field F2.We study the enumeration and Hamming weight distribution of the repeated-root constacyclic codes over Fpm+uFpm+vFpm+uvFpm.The structure and some related problems of constacyclic codes over finite non-chain rings Fq+vFq+uFq+uvFq are also studied. The details are given as follows:(1)We give a set of generators of (1+u)-constacyclic codes of an arbitrary length over the finite non-chain ring F2+uF2+vF2.A Gray map is defined,We prove that the Gray image of a linear(1+u)-cyclic code over F2+uF2+vF2is a binary distance invariant linear quasi-cyclic code. A number of optimal binary linear quasi-cyclic codes are obtained.We also prove that,if n is odd,then every binary code which is the Gray image of a linear cyclic code over F2+uF2+vF2is equivalent to a quasi-cyclic code of order two.(2)All (1+u)-constacyclic codes of length2k over the finite non-chain ring F2+uF2+vF2are classified,and a formula for the number of these constacyclic codes is obtained.Furthermore,all the dual codes of (1+u)-constacyclic codes are obtained.(3)We classify all the (α+βu)-constacyclic codes of length pe over the finite non-chain ring Fpm+uFpm+vFpm+uvFpm,where u2=0, v2=0,uv=vu,α,β∈F*pm.For any given α,β,e, a formula for the number of these constacyclic codes is obtained. Then,the Torsion code and Residue code of those constacyclic codes are also defined.Finally,we obtain the Hamming distance distribution of those constacyclic codes with the help of their Torsion codes.(4) We introduce the finite non-chain ring Fq+vFq+uFq+uvFq, where v2=v,u2=u, uv=vu,q=pm, p is a prime, and study constacyclic codes over this ring. It is shown that all constacyclic codes over this ring of arbitrary lengths are principally generated. We also give the dual codes of these constacyclic codes. A Gray map from (Fq+vFq+uFq+uvFq) to Fq4n is introduced and Gray images of constacyclic codes over Fq+vFq+uFq+uvFq are also considered. Finally, some optimal q-ary codes are obtained as the images of constacyclic codes over Fq+vFq+uFq+uvFq under Gray map. Furthermore, if the length and the character are coprime, then some classes of constacyclic codes over F+vFq+uF+uvFq must contain a unique idempotent generator polynomial.