Random Construction of Partial MDS Codes

TL;DR

This paper extends the construction of partial MDS codes using maximum rank distance codes, provides an algebraic framework for their generator matrices, and shows that random matrices can generate PMDS codes with high probability over large fields.

Contribution

It generalizes known constructions of PMDS codes to any MRD code and introduces an algebraic standard form for their generator matrices.

Findings

Random generator matrices in standard form produce PMDS codes with high probability over large fields.

Provides algebraic conditions for the existence of PMDS codes.

Extends construction methods to any MRD code, not just Gabidulin codes.

Abstract

This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.