Parity check systems of nonlinear codes over finite commutative Frobenius rings

TL;DR

This paper extends the concept of parity check systems from linear binary codes to nonlinear codes over finite commutative Frobenius rings, providing new insights into their properties and connections to characters.

Contribution

It introduces parity check systems for nonlinear codes over Frobenius rings and explores their properties, including a MacWilliams type theorem on distance distribution.

Findings

Fundamental properties of nonlinear codes over Frobenius rings derived

Connection between parity check systems and characters established

MacWilliams type theorem for these codes proved

Abstract

The concept of parity check matrices of linear binary codes has been extended by Heden [9] to parity check systems of nonlinear binary codes. In the present paper we extend this concept to parity check systems of nonlinear codes over finite commutative Frobenius rings. Using parity check systems, results on how to get some fundamental properties of the codes are given. Moreover, parity check systems and its connection to characters is investigated and a MacWilliams type theorem on the distance distribution is given.