Low-Rank Parity-Check Codes Over Finite Commutative Rings

TL;DR

This paper introduces Low-Rank Parity-Check (LRPC) codes over finite commutative rings, providing new theoretical bounds and extending their construction beyond finite fields, with implications for network coding and cryptography.

Contribution

It defines LRPC codes over finite commutative local rings, improves decoder failure probability bounds, and extends code construction to arbitrary finite rings without Galois extension degree restrictions.

Findings

Improved theoretical failure probability bounds for LRPC decoders

Construction of LRPC codes over arbitrary finite commutative rings

Elimination of the need for prime extension degree in code construction

Abstract

Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension.