The Weight Distributions Of Several Cyclic Codes Over Finite Fields

In this paper,we mainly study several reducible cyclic codes over finite fields.The details are presented as follows:(1)Firstly,two classes of cyclic codes over Fp whose duals have two zeros with few weights are constructed,where p is an odd prime.Based on the known results of exponential sums and quadratic forms,the weight distributions of these codes are determined.Some of the cyclic codes are optimal,which refers to meeting a certain bound on linear codes.Besides,the number of nonzero weights of these codes is no more than five and these codes can be employed to obtain secret sharing schemes,authentication codes and association schemes.(2)Let Fqm1 and Fqm2 denote two finite fields with qm1 and qm2 elements,respectively,where m1,m2,are two distinct positive integers such that gcd(m1,m2)=1.the cyclic codes C(e1,e2)over Fq whose duals have two zeros are constructed,where e1,e2 are two positive integers.Based on the known results of Gauss sums over Fq,the codes C(e1,e2)are shown to have at most five nonzero Hamming weights,which contain two subclasses of three-weight optimal cyclic codes according to Griesmer bound.The optimal three-weight cyclic codes with m1=1 and gcd(e2,qm2-1/q-1)=1 have been constructed in[45]by using different method.Moreover,some duals of C(e1,e2)are also studied.