Some Classes Of Negacyclic Codes Over Finite Fields

Negacyclic codes over finite fields are an important class of linear codes.Due to good algebra structure,negacyclic codes over finite fields have many applications in consumer electronics,data transmission technologies and communication system.Hence,negecyclic codes over finite fields have always been an importanct topic in coding theory.In this paper,the structure and performance of negacyclic codes over the finite field F_q are studied,and some optimal negacyclic codes are obtained.There are the following two parts in this paper.(1)The minimum distance of negacyclic codes over F_q of length n=(q~m-1)/2 with two zeros are explored.Sufficient and necessary conditions on the minimum distance of such negacyclic codes are given.It is shown that such negacyclic codes can produce optimal linear codes only when q=3 or 5.Six classes of optimal quinary negacyclic codes with parameters[(5~m-1)/2,(5~m-1)/2-2m,4]are constructed by using some almost perfect nonlinear monomials and other monomials.Under the condition that m is even,a comparison with the cyclic codes shows that optimal linear codes can be obtained from such negacyclic codes.(2)Ternary linear complementary dual(LCD)BCH negacyclic codes of length(3~m-1)/2 are studied.By using the structure and distribution of cyclotomic cosets,the dimension of ternary negacyclic LCD BCH codes are determined.The minimum distance of them are analyzed by the BCH bound for negacyclic codes.Some optimal ternary negacyclic LCD codes are derived with the help of Magma.