Weight Distributions Of Some Cyclic Codes And Linear Codes

Linear codes,especially cyclic codes,play a very significant role in the theory of error-correcting codes.Cyclic codes can be efficiently encoded using shift registers,and they have rich algebraic structures for efficient error detection and correction.In coding theory it is often desirable to know the weight distribution of the codes because they can be used to estimate the error correcting capability and the error probability of error detection with respect to some decoding algorithms.This is quite useful in practice.Therefore,determining the weight distributions of some linear codes and cyclic codes is not only a problem of theoretical interest,but also of practical importance.Weight distributions of some cyclic codes and linear codes are determined in this paper.The main content of this paper is organized as follows.In Chapter 1,first we introduce the research background and the known result,and then we list the main result in this paper.Chapter 2 reviews some elementary results that will be used in this paper.In Chapter 3,let m,k and d be positive integers such that 2?k??m+1?/2 and gcd ?m,d?=1.Let p be an odd prime and?be a primitive element of the finite field F_pm.For j=0,1,···,k-1,let h_j?x? and h_-j?x? be the minimal polynomials of ?^-?pjd+1?/2 and -?^-?pjd+1?/2 over F_p,respectively.Let C_m,d,2k and C_m,d,2k-1 be the cyclic codes with parity-check polynomial ?_j=0^k-1h_j?x?h_-j?x? and h_k-1?x??_j=0^k-2h_j?x?h_-j?x?,respectively.In this paper,the weight enumerator of the cyclic code C_m,d,2k is determined,and the weight enumerator of the cyclic code C_m,d,2k-1 is also determined in the condition of 2???m/?gcd?m,k-1??.In Chapter 4,first we construct some linear codes by finite sets over finite fields,which are determined by quadratic forms,then we show the weight distribution of these linear codes.