Research On Algebraic Theory Of Several Types Of Error-correcting Codes

Linear codes are the main subject of error-correcting codes since they have com-position performance which contributes to the operation analysis.The study of linear codes has attracted many researchers for a long time and it has become an active re-search branch of the theory of error-correcting codes.In this thesis,by an application on number theory and the theory of finite fields or even finite rings in algebra,especially by the methods and tools of cyclotomic cosets and irreducible factorization of polynomials,as well as the ideas of homomorphisms of rings and the Chinese remainder theorem,we take an intensive study on several kinds of important linear codes over finite fields and finite rings,such as constacyclic codes,symbol-pair codes and subspace codes,and obtain the following four aspects results about the structures and the fundamental parameters of those codes.At first,we carry out the research on the constacyclic codes on finite fields with characteristic p.This research extends the length of constacyclic codes.The irreducible factorization of xklmpn-?(??Fq*)over Fq in four cases according to the division relations of the prime number k,l and q-1 are provided.Therefore,the structures of constacyclic codes of length klmpn over a finite field Fq with characteristic p are obtained for different odd primes k,l and p.The former results correct two errors of Tong's paper in 2016.We next consider all the ?+u?-constacyclic codes of length nps over R,where R=Fpm+uFpm with u2=0,?,??Fpm*,n,s? N+and gcd(n,p)=1.Let ?0?Fpm*satisfying ?0ps=?.Then the residue ring R[x]/<xnps-?-u?>is a chain ring with the maximal ideal<xn-?0>in the case that xn-?0 is irreducible in Fpm[x].If xn-?0 is reducible in Fpm[x],we give the explicit expressions of the ideals of R[x]/<xnps-?-u?)by using the irreducible factorization of xn-?0 in Fpm[x].Besides,the size of each?+u?-constacyclic code is provided and the generator polynomial and the size of the dual code of an ?+u?-constacyclic code is discussed.Symbol-pair codes are proposed to protect against pair-errors in symbol-pair read channels.Being a typical type of error-correcting codes,just as usual,the higher the minimal pair distance,the more pair errors the symbol-pair codes can correct.Maximal distance separable(MDS)symbol-pair codes are optimal in the sense that the minimal pair distance cannot improved for a given length and a code size.In this dissertation,we give some lower bounds for minimal pair distance of repeated-root constacyclic codes.Besides,we obtain new MDS symbol-pair codes with infinity length and other MDS symbol-pair codes with minimal pair distance 6.Subspace codes and particularly cyclic subspace codes have attracted a lot of re-searchs due to their applications in error correction for random network coding.However,the study of this kind of codes is not intensive enough.We explore the parameters of a class of the cyclic subspace codes whose subspace polynomials are more generalized.Therefore,some results as the corollaries of these acquired parameters are obtained and they extend the previously known results.Finally,a construction of k-dimensional cyclic subspace codes of size qN-1/q-1 and minimal distance 2k-2 is provided.These results enrich the relevant theories of linear codes,and they will help us to grasp,understand and apply these linear codes more deeply.