Several Classes Of Linear Codes And Their Parameters

Linear codes are an important class of error correcting codes.Cyclic codes are a subclass of linear codes.Because of its good algebraic structure and efficient decoding algorithm,cyclic codes have important applications in the fields of communication,cryptography,data storage and so on.The parameters of linear codes mainly include length,dimension,and minimum distance,and the three parameters have a certain system.The construction of linear codes with parameters reaching the theoretical bound is a hot issue in the field of error correcting codes.Linear complementary dual(LCD)codes and self-orthogonal codes are two special linear codes.Linear complementary dual(LCD)codes could be used in cryptography to resist side-channel attacks and fault non-invasive attacks.Self-orthogonal codes can be used to construct quantum error-correcting codes,which can protect quantum information in quantum computations and quantum communications.First,we construct a family of cyclic codes over Fp of length pm-1/p-1 with generator polynomial mu(x)mv(x),where p is an arbitrary odd prime,m is a positive integer with gcd(m,p-1)= 1 and(u,v)=(p-1,2(p-1)).This family of cyclic codes is shown to have parameters[pm-1/p-1,pm-1/p-1-2m,d],4?d?5,which is at least almost optimal.A generic construction of linear codes over Fp from subsets of Fpm was considered by Ding.In the following,we restate this generic construction from the view point of vector space.we construct four classes of linear codes with the generic construction introduced before by choosing four types of defining sets.We give very simple characterizations of these linear codes to be LCD codes or self-orthogonal codes.Using the characterizations,we obtain infinite families of binary LCD codes and self-orthogonal codes from the proposed linear codes and their duals.Many LCD codes and self-orthogonal codes presented in this paper are optimal or almost optimal codes.Thanks to some properties of Krawtchouk polynomials,we will determine the weight distributions of two sub-families of these linear codes.