Several Classes Of Cyclic Codes Over Finite Fields

Cyclic codes are an important class of error-correcting codes and have good algebraic structure.Since encoding and decoding circuits are easily implemented,cyclic codes over finite fields have widely been used in many areas.When length and dimension are fixed,one main theme in the theory of cyclic codes is to seek cyclic codes with maximum minimum distance.Assume that m is an integer greater than 2,p is an odd prime number and n=2(pm-1)/(p-1).In this dissertation,cyclic cods over the finite field IFp with two zeros or three zeros are studied.By combining the sphere packing bound p-ary[n,n-2m,4]and[n,n-3m/2-1,4]cyclic codes are obtained.Specific results are listed as follows:(1)Cyclic codes with two zeros.A sufficient condition that such cyclic codes have minimum distance not less than 3 is given.Then it is proved that a p-ary linear code with parameters[n,n-2m,4]is optimal.By analyzing the existence of solutions of some equations over Fnm,three classes of optimal cyclic codes with parameters[n,n-2m,4]are constructed.Based on Delsarte theorem and some known exponential sums,the weight distribution of the duals of two classes of cyclic codes is determined.The dual codes are shown to have only three nonzero weights or seven nonzero weights.(2)Cyclic codes with three zeros.For any even number m?2,it is proved that a p-ary linear code with parameters is optiimal.A sufficient codition for cyclotomic cosets of p modulo n to have size m/2 is given.By choosing appropriate cyclotomic cosets and zeros,four classes of optimal cyclic codes with parameters[n,n-3m/2-1,4]are constructed.