Construction of Z4-linear Reed-Muller codes

TL;DR

This paper introduces new quaternary constructions to generate Z4-linear Reed-Muller codes that mirror the properties of classical binary Reed-Muller codes, expanding the coding theory landscape.

Contribution

It presents novel quaternary Plotkin constructions and demonstrates their use in creating new Reed-Muller code families with preserved parameters after Gray mapping.

Findings

Constructed new quaternary Reed-Muller codes with similar parameters to binary counterparts.

Established duality relationships using the Kronecker inner product.

Analyzed code parameters such as length, dimension, and minimum distance.

Abstract

New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear Reed-Muller family. To make more evident the duality relationships in the constructed families the concept of Kronecker inner product is introduced.