MDS矩阵和对合MDS矩阵的新构造方法

首先对Lacan等人给出的由Vandermonde矩阵构造MDS码的方法进行了研究, 指出了其中存在的问题, 给出了由两个Vandermonde矩阵构造MDS矩阵的充要条件; 然后利用矩阵乘的方法, 给出了由标量乘Vandermonde矩阵构造MDS矩阵的充要条件; 最后在Sajadieh等人给出的由两个Vandermonde矩阵构造对合MDS矩阵方法的基础之上, 给出了标量乘Vandermonde矩阵构造对合MDS矩阵的方法。对标量乘矩阵来讲, 可以通过调控标量中分量的大小来调整标量乘矩阵元素大小和元素重量大小来满足其软、硬件实现性能, 因此该构造MDS矩阵及对合MDS矩阵的方法具有实用价值。

New construction methods for MDS matrices and involution MDS matrices

Firstly this paper studied the method of constructing MDS codes by Vandermonde matrices proposed by Lacan et al and point out the problems existing in this method, and proposed the necessary and sufficient conditions of constructing MDS matrices by two Vandermonde matrices. Then, using the method of matrix multiplication, this paper proposed the necessary and sufficient conditions of constructing MDS matrices by scalar multiplication Vandermonde matrices. Finally, based on the method of constructing involution MDS matrices from two Vandermonde matrices proposed by Sajadieh et al, this paper proposed the method of constructing involution MDS matrices by scalar multiplication Vandermonde matrices. For scalar multiplication matrices, it could adjust elements size and weight in scalar multiplication matrices through regulating the size of scalar components to meet the implementation performance of software and hardware. So the methods of constructing MDS matrices and involution MDS matrices have practical value.