New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code

TL;DR

This paper introduces five new infinite families of linear near-MDS codes that are uniformly packed and serve as almost perfect multiple coverings, constructed via m-lifting of codes based on the ternary Golay code.

Contribution

The paper presents a novel construction method for infinite families of near-MDS codes using m-lifting, expanding the class of codes with specific packing and covering properties.

Findings

Five new infinite families of UPWS near-MDS codes

Codes are almost perfect multiple coverings of deep holes

Construction based on m-lifting of ternary Golay code

Abstract

We present five new infinite families of linear near-MDS codes uniformly packed in the wide sense (UPWS). These codes are also almost perfect multiple coverings of the deep holes or farthest-off points (APMCF), i.e.\ the vectors lying at distance $R$ (covering radius) from the code. The families are constructed by $m$-lifting when one takes a starting code $C$ over the ground Galois field $\F_q$ with a parity check matrix $H(C)$ and then considers the codes $C_m$ over $F_{q^m}$, $m\ge2$, with the same parity check matrix $H(C)$. As starting codes we used the ternary perfect Golay code and codes obtained by its extension and puncturing. To prove the needed combinatorial properties (UPWS and APMCF), we used the $m$-lifting of the dual codes and features of near-MDS codes. A general theorem on infinite families of UPWS near-MDS codes is proved.