Weight Distributions Of Three Classes Of Cyclic Codes

Cyclic codes are a subclass of linear block codes. They can be easily designed and have good algebraic structure that are convenient for analysis. Besides, they can be effi-ciently encoded using shift registers technically. Moreover, they have efficient encoding and decoding algorithms. So cyclic codes have wide applications in communication and storage systems. The weight distribution of a cyclic code gives the minimum distance of it and, thus, the error correcting capability. In addition, the weight distribution can be used to estimate the error probability of error detection and correction with respect to some decoding algorithms. Therefore, determining the weight distributions of cyclic codes is not only a problem of theoretical interest, but also of practical importance. There are many remarkable results on the weight distributions of cyclic codes. Inspired by these original ideas, three classes of reducible cyclic codes are constructed and their weight distributions are determined in this paper. The main content of this paper can be generalized as the following.In Chapter one, we introduce the research backgrounds, some results on the weight distributions of cyclic codes, and then list the main work and some preliminaries.In Chapter two, let m be an odd positive integer and k be any positive integer such that s=m/d≥3, where d=gcd(m, k). Let t be a positive divisor of d and m0=m/t. Under these conditions, a class of reducible cyclic codes C1 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-1, (-π)-1 and π-(p+1)/2 over Fpt, where p is an odd prime and π is a primitive element of m. By calculation, C1 is a cyclic code over Fpt with parameters [pm 1,3m0,2-pt-1/2pm-t]. Besides, the weight distribution of C1 is determined.In Chapter three, let m and k be any two positive integers such that s=m/d≥5 is odd, where d=gcd(m,k). Let t be a positive divisor of d such that d/t is odd, m0=m/t. Let p and π be defined as above. Then a class of reducible cyclic codes C2 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-2,π-(pk+1) and π-(p2k+1) over Fpt Moreover, C2 is proved to be a cyclic code over Fpt with parameters [pm-1,3m0,(pt-1)(pm-t-pm+3d-2t)]. Furthermore, the weight distribution of C2 is determined.In Chapter four, let m, k, d, t, m0, p and π be defined as in Chapter three. A class of reducible cyclic codes C3 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-1, π-2,π-(pk+1) and π-(p2k+1) over Fpt. Moreover, in this chapter, C3 is proved to be a cyclic code over Fpt with parameters [pm-1,4m0,(pt-1)pm-t-pm+4d-t/2]. Furthermore,the weight distribution of C3 is determined.