The Study For Self-Dual Cyclic-Codes

Based on the effective encoding and decoding algorithms,cyclic codes do not only play an important role in Error-Correction code theory,but also have a wide range of applications in the field of communications.Polynomial rings and ideals are usually used to construct cyclic codes.The skew polynomial ring is a non-commutative ring,which is widely applied in the construction of the algebraic codes.Because the polynomial in the skew polynomial rings do not satisfy the commutative law of multiplication,so there are more ideals in the skew polynomial rings than in the polynomial rings.In[3-5],a new form of codes are constructed by using skew polynomials,which are called skew cyclic codes,and many good codes are constructed.In this thesis,we studied skew cyclic codes and self-dual skew cyclic codes.At first,the bivariate skew polynomial on the ring Z2 +uZ2 + u2Z2 is discussed and its factorized form is given.The properties and construction of the 2-D skew cyclic code are discussed.The necessary and sufficient conditions for the linear codes to be 2-D skew cyclic codes are studied.As a special linear code,self-dual codes have been widely applied in the theory of error correcting codes.There are many construction methods of self-dual codes.One of the ideal methods is to construct a larger self-dual code with smaller length of self-dual code.This method is called construction method.In this paper,we aim to extend the self-dual codes to self-dual module ?-constacyclic cyclic codes.We discuss the module ?-constacyclic cyclic codes over Fpm+vFpm(v2 = v);we give the necessary and sufficient conditions for the self-dual module ?-constacyclic cyclic codes by direct sums.In particular,we discuss the self-dual module ?-constacyclic cyclic codes over F5m+v5m,and prove that there are some self-dual cyclic codes of auto-isomorphism mapping ?.This paper also studied the problem of counting the self-dual quasi-cyclic codes.In[32],the QC codes are calculated by decomposing the generalized quasi cyclic code(QC code)into straight sum of irreducible circulate block,and the reasoning process and complexity are both high.In[42,43],using the Chinese remainder theorem to decompose the QC code,the problem of counting QC codes after decomposition is simplified to simplify the complexity.We have learned and popularized the idea of this decomposition,we regard the quasi-twist code as a linear code on the ring.When m is coprime with the characteristics of Fq,we use the Chinese Remainder Theorem and discrete Fourier transform to decompose rings Fq[x]/(xm+1)into direct products on domains.Therefore,the quasi negative cyclic codes can be decomposed into low growth codes,and further study them systematically.Finally,the counting formula of self dual quasi negative cyclic codes with the length of l2a and the subscript l on the Fq is given.