On linear lexicographic codes: Ninth column construction of the ternary Golay code

TL;DR

This paper characterizes linear lexicographic p-ary codes and determines their dimensions for various bases, revealing that the ternary Golay code can be constructed as a ninth column code with specific parameters.

Contribution

It provides a new characterization of linear lexicographic codes and identifies conditions under which higher-dimensional codes, including the ternary Golay code, can be obtained.

Findings

Dimensions of lexicographic codes are determined for p ≥ 3.

The ternary Golay code is obtained as a ninth column code for p=3, d=6.

Higher-dimensional codes are possible in certain minimum distance cases.

Abstract

We characterize linear lexicographic $p$-ary codes. Using this characterization, when $p \ge 3$, we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain minimum distances. In these excluded cases, we may obtain linear codes of higher dimensions; for instance, when $p = 3$ and $d = 6$, the ternary Golay code is obtained.