مرکز منطقه ای اطلاع رساني علوم و فناوري - On generator matrices of MDS codes (Corresp.)

Abstract :

It is shown that the family of $q$ -ary generalized Reed-Solomon codes is identical to the family of $q$ -ary linear codes generated by matrices of the form $[I|A]$ , where $I$ is the identity matrix, and $A$ is a generalized Cauchy matrix. Using Cauchy matrices, a construction is shown of maximal triangular arrays over GF $(q)$ , which are constant along diagonals in a Hankel matrix fashion, and with the property that every square subarray is a nonsingular matrix. By taking rectangular subarrays of the described triangles, it is possible to construct generator matrices $[I|A]$ of maximum distance separable codes, where $A$ is a Hankel matrix. The parameters of the codes are $(n,k,d)$ , for $1 \\leq n \\leq q+ 1, 1 \\leq k \\leq n$ , and $d=n-k+1$ .