A Construction of Matrices with No Singular Square Submatrices

This paper presents a new construction of matrices with no singular square submatrix. This construction allows designing erasure codes over finite fied with fast encoding and decoding algorithms. 1 Systematic MDS Erasure Codes It is well known that a [n, k, d]-error correcting code is Maximum Distance Separable (MDS) if and only if its k × n-generator matrix does not contain any singular k×k-square submatrix [1, p. 319, Cor. 3]. Similarly, a MDS code is systematic if and only if its generator matrix has the form (Ik|R), where Ik is the k× k-identity matrix and R is a (k× (n− k))-matrix such that any r× r-square submatrix is nonsingular [1, p. 321, Th. 8], for r ≤ k. It should be noted that systematic MDS codes built from this property (i.e. from the matrix R) are used in practical computer communications to cope with losses of data packets [2]. Cauchy matrices are generally used to build matrices over finite fields whose any square submatrix is non-singular [3]. A r × r-Cauchy matrix is defined as ( 1 ai−bj ) r−1 i,j=0, where (ai) r−1 i=0, and (bj) r−1 j=0, are given vectors of (IFq) . Such a matrix is nonsingular if and only if the elements ai, i = 1, . . . , r are distinct, the elements bj , j = 1, . . . , r are distinct and ai + bj 6= 0, 1 ≤ i ≤ j ≤ j. It can be easily verified that any submatrix of a Cauchy matrix is a Cauchy matrix, and then any square submatrix of a nonsingular Cauchy matrix is nonsingular. It should be noted that the Vandermonde matrices defined over finite field can contain singular square submatrices [1, p. 323, Problem 7]. 2 A New Class of Matrices with No Singular Square Submatrices Theorem 1. Let us denote by A and B two r×r-matrices of rank r over a given field such that any r×r-submatrix of the r×2r-matrix (A|B) has a rank r. Then, the matrix A−1B is such that any of its square submatrices is nonsingular. Proof. Let us denote byW the r×2r-matrix (A|B). The product A−1W is of the form (Ir|C), where C = A−1B. Since, by construction, any r × r-submatrix of W is nonsingular then any r×r-submatrix of the product A−1W is nonsingular. By combining [1, p. 319, Cor. 3] and [1, p. 321, Th. 8], it can be stated that a r × 2r-matrix on the form [Ir|C] whose any r × r-matrix is nonsingular is necessarily such that C does not contain any r′ × r′-singular submatrix for r′ ≤ r. This concludes the proof. Note that it can be verified that these matrices are not Cauchy matrices. Let us give a counterexample. Let us work in the field IF5 and let us consider the matrices A and B respectively equal to ( 1 1 1 2 ) and ( 1 1 3 4 ) . Then, the product A−1 ×B is equal to ( 2 4 4 1 ) × ( 1 1 3 4 ) = ( 4 3 2 3 ) . From the definition of Cauchy matrices, it can be easily verified that this product cannot be a Cauchy matrix. The construction was presented with square matrices, but it can be generalized when A is a k×k-matrix and B is a k× (n−k) matrix. One can then build a k× (n− k) matrix , for any ”suitable” n (n ≥ k), whose any square submatrix is nonsingular. Such matrix can be directly used to build the generator matrix of a systematic MDS codes (see Section 1). 3 Application of this construction to build fast erasure codes In order to apply this construction to design efficient erasure codes for computer communications, one must consider matrices for which there exist fast matrixvector multiplication and fast inversion algorithms. The Vandermonde matrices have these properties. These matrices are defined from a vector of r distinct elements (a1, . . . , ar) of (IFq) as ( a i )r