Constructions of MDS codes via random Vandermonde and Cauchy matrices over small fields

Vandermonde and Cauchy matrices are commonly used in the constructions of maximum distance separable (MDS) codes. However, when additional design constraints are imposed on the code construction in addition to the MDS requirement, a Vandermonde or Cauchy matrix may not always suffice. We discuss some related coding problems of that nature that arise in different practical settings. We present a useful technique to tackle the constrained coding problems that includes random selection of the evaluation points of a Vandermonde or a Cauchy matrix. Our solutions require small finite fields whose sizes are polynomial in the dimensions of the generator matrices. We believe that this technique will be useful for solving a broad range of coding problems.