Properties of Maximally Recoverable Product Codes and Higher Order MDS Codes

Abstract

Product codes are a class of codes which have generator matrices as the tensor product of the component codes and the codeword itself can be represented as an (m × n) array, where the component codes themselves are referred to as the row and column codes. Maximally recoverable product codes (MRPCs) are a class of codes which can recover from all information theoretically recoverable erasure patterns, given the $a$ column and $b$ row constraints imposed by the code. In this work, we derive puncturing and shortening properties of maximally recoverable product codes. We give a sufficient condition to characterize a certain subclass of erasure patterns as correctable and another necessary condition to characterize another subclass of erasure patterns as not correctable. In an earlier work, higher order MDS codes denoted by MDS(l) have been defined in terms of generic matrices and these codes have been shown to be constituent row codes for maximally recoverable product codes for the case of $a$ = 1. We derive a certain inclusion-exclusion type principle for characterizing the dimension of intersection spaces of generic matrices. Applying this, we formally derive a relation between MDS(3) codes and points/lines of the associated projective space.