秩距离缩短码的构造

The Construction of Rank Distance Abridging Codes

Gabidulin提出了秩距离码及最大秩距离码的理论 ,给出了判断码的最小秩距离的方法 ,并通过引进线性化多项式的概念 (类似于纠错码 )构造了一些最大秩距离码 ,并对这些最大秩距离码进行了分类 ,其中包括线性 q-循环码和最大秩距离 Reed- Solomon码 .该文在此基础上提出了秩距离缩短循环码、秩距离缩短 Reed- Solomon码以及秩距离缩短 BCH码的概念 (类似于纠错码 ) ,给出了秩距离缩短循环码的生成矩阵和校验矩阵 ,给出了秩距离缩短 Reed- Solomon码以及秩距离缩短 BCH码的校验矩阵 。

The theory of rank distance codes and maximum rank distance codes was introduced by Gabidulin E.M. The method of determining the minimum rank distance of a given code was given also. By introducing the concept of linearized polynomial, similar to error correcting codes, some kinds of maximum rank distance codes were constructed. These maximum rank distance codes were classified, therein including linear q cyclical codes, and maximum rank distance Reed Solomon codes. For general error correcting codes, the abridging cyclic codes can be constructed as follows. Given a cyclic code, by taking those codewords whose information digits are zero in the front positions from 1 to i as codewords of the new constructed code, a (1ik) code is obtained. The new constructed code is referred to as abridging code of the original cyclic code, referred to as abridging cyclic code for short. On the basis of these, similar to error correcting codes, the concepts of rank distance abridging cyclic codes and rank distance abridging Reed Solomon codes as well as rank distance abridging BCH codes are put forward in this paper. These codes are obtained from linear rank distance q cyclical codes and rank distance Reed Solomon codes as well as rank distance BCH codes respectively, by taking those codewords whose information digits are zero in the front positions from 1 to i as codewords of the new constructed code. The generator matrix and parity check matrix of rank distance abridging cyclic code are given in this paper. The generator matrix can be found out by means of the generator polynomial or can be obtained by deleting the front rows and columns from 1 to i from the generator matrix of the original linear rank distance q cyclical code, while the parity check matrix can be found out by means of the parity check polynomial or can be obtained by deleting the front columns from 1 to i from the parity check matrix of the original linear rank distance q cyclical code. Besides these, the parity check matrices of rank distance abridging Reed Solomon codes and rank distance abridging BCH codes are given. It is proved that the rank distance abridging Reed Solomon codes and rank distance abridging BCH codes can constitute maximum rank distance codes. The minimum rank distances of them are found out.