In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Rema...In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.展开更多
The weight hierarchy of a [n, k; q] linear code C over Fq is the sequence （d1,…, dr,… , dk）, where dr is the smallest support weight of an r-dimensional subcode of C. In this paper, by using the finite projective ...The weight hierarchy of a [n, k; q] linear code C over Fq is the sequence （d1,…, dr,… , dk）, where dr is the smallest support weight of an r-dimensional subcode of C. In this paper, by using the finite projective geometry method, we research a class of weight hierarchy of linear codes with dimension 5. We first find some new pre- conditions of this class. Then we divide its weight hierarchies into six subclasses, and research one subclass to determine nearly all the weight hierarchies of this subclass of weight hierarchies of linear codes with dimension 5.展开更多
当 A是特殊的 (n,1,n)等重码和 (n,n,1)等重等距码时 ,通过对两个旧码 A和 B构造一个新码 A B即直和码的分析 ,给出了 A B广义 Ham ming重量与 A、B的广义 Ham ming重量的关系 ;在 A、B是一般情形下 ,对直和码的广义 Ham ming重量进行...当 A是特殊的 (n,1,n)等重码和 (n,n,1)等重等距码时 ,通过对两个旧码 A和 B构造一个新码 A B即直和码的分析 ,给出了 A B广义 Ham ming重量与 A、B的广义 Ham ming重量的关系 ;在 A、B是一般情形下 ,对直和码的广义 Ham ming重量进行了研究。展开更多
Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goet...Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goethals codes and their dual codes are proved to be cyclic.展开更多
文摘In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.
基金supported by the National Natural Science Foundation of China (Nos. 61303212 and 61170080)the State Key Program of the National Natural Science of China (Nos. 61332019 and U1135004)the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (No. CZY14019)
文摘The weight hierarchy of a [n, k; q] linear code C over Fq is the sequence （d1,…, dr,… , dk）, where dr is the smallest support weight of an r-dimensional subcode of C. In this paper, by using the finite projective geometry method, we research a class of weight hierarchy of linear codes with dimension 5. We first find some new pre- conditions of this class. Then we divide its weight hierarchies into six subclasses, and research one subclass to determine nearly all the weight hierarchies of this subclass of weight hierarchies of linear codes with dimension 5.
文摘Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goethals codes and their dual codes are proved to be cyclic.
