Linear recurring sequences and subfield subcodes of cyclic codes  

作　　者：GAO ZhiHan FU FangWei 

机构地区：[1]Department of Combinatorics and Optimization, University of Waterloo [2]Chern Institute of Mathematics and LPMC, Nankai University

出　　处：《Science China Mathematics》2013年第7期1413-1420,共8页中国科学：数学（英文版）

基　　金：supported by National Key Basic Research Program of China(973 Program)(Grant No.2013CB834204);National Natural Science Foundation of China(Grant Nos.61171082 and 10990011)

摘　　要：Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let M qm （f（x）） denote the set of all linear recurring sequences over F qm with characteristic polynomial f（x） over F q m. Denote the restriction of M qm （f（x）） to sequences over Fq and the set after applying trace function to each sequence in M qm （f（x）） by M q m （f（x）） ｜Fq and Tr（M qm （f（x）））, respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over F qm .

关 键 词：linear recurring sequences characteristic polynomial cyclic codes subfield subcodes trace codes 

分 类 号：O157.4[理学—数学] S511.03[理学—基础数学]