On The Algebraic Structures Of Several Types Of Additive Codes And Double Cyclic Codes

Additive codes and double cyclic codes are important code classes in error correcting codes,and they have good algebraic structure.There are important applications on the construction of linear codes,the construction of quantum error-correcting codes and the construction of covering codes.This paper carries out in-depth research on algebraic structures of Z _pZ _p[u]-additive cyclic codes,Z _pZ _p[v]-additive cyclic codes and F_{q 1} F_q2-additive double cyclic codes,and some good results are obtained in their applications.The details are given as follows:The algebraic structure of Z _pZ _p[u]-additive cyclic codes and their dual codes are studied.First of all,The definitions,generator polynomials and minimal generator sets of Z _pZ _p[u]-additive cyclic codes are given.Then the generator polynomials of dual codes are determined.Finally,as an application,we construct some quantum codes from Z₂ Z₂[u]-additive cyclic codes.The algebraic structure of Z _pZ _p[v]-additive cyclic codes and their dual codes are studied.First of all,The definitions,generator polynomials and minimal generator sets of Z _pZ _p[v]-additive cyclic codes are given.Then the generator polynomials of dual codes are determined.Meanwhile,we give some optimalZ _pZ _p[v]-linear codes and MDSS codes.Finally,as an application,we construct some quantum codes fromZ _pZ _p[v]-additive cyclic codes.The algebraic structure of F_{q 1} F_q2-additive double cyclic codes and their dual codes are studied.First of all,The definitions,generator polynomials and minimal generator sets of F_{q 1} F_q2-additive double cyclic codes are given.Then the generator polynomials of dual codes are determined.Finally,as an application,we list some examples of best-know or optimal F₂ F₃-additive double cyclic covering codes.