The Construction Of Self-orthogonal Codes And Its Applications

Self-orthogonal codes over finite fields or rings have been an important subject in the research of error-correcting codes.With a deeply study in quantum error-correction,it has been found that classical self-orthogonal codes can be used construct quantum error-correcting codes.This has caused the great interest in constructing classical self-orthogonal codes.In this paper,we study the structure of cyclic self-orthogonal codes over the finite ring F_q+uF_q,where u²=0,and construct cyclic self-orthogonal codes over F_q+uF_q.Based on such self-orthogonal codes,quantum codes with good parameters are obtained.Meanwhile,by using Hermitian self-orthogonal constacyclic codes over F_4^m,quantum maximal distance separate（MDS）codes are constructed.The paper includes three parts:1.Cyclic self-orthogonal codes over F₂+uF₂ are studied.A generator polynomial of any cyclic self-orthogonal code over F₂+uF₂ is given.All cyclic self-orthogonal codes overF₂+uF₂ of odd length are obtained.2.The structure of Hermitian self-orthogonal constacyclic codes over F_4^m of any length is investigated.By using generator polynomial,the existence of Hermitian self-orthogonal constacyclic codes over F_4^m is explored and the enumeration formula of such codes is determined.Quantum MDS codes are contructed from Hermitian self-orthogonal constacyclic codes over F_4^m.3.A sufficient and necessary condition for the existence of cyclic self-orthogonal codes over F_q+uF_q is given,where q（?）1（mod 4）.A Gray map fromF_q+uF_q to F_q² is introduced.Self-orthogonal codes over qF are obtained from cyclic self-orthogonal codes over F_q+uF_q under this Gray map.Further,quantum codes with good parameters are constructed by using the Steane’s enlargement construction.