A generalized concatenation construction for q-ary 1-perfect codes

TL;DR

This paper introduces a generalized concatenation method for constructing q-ary 1-perfect codes of specific lengths, extending existing codes and providing a q-ary analogue of Phelps construction.

Contribution

A new generalized concatenation construction for q-ary 1-perfect codes that expands the range of constructible code lengths and parameters.

Findings

Constructs q-ary 1-perfect codes of length (q - 1)nm + n + m.

Enables creation of codes with parameters (q^{s_1 + s_2}, q^{q^{s_1 + s_2} - (s_1 + s_2) - 1}, 3)_q.

Provides a q-ary analogue of Phelps construction.

Abstract

We consider perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). In this paper, a generalized concatenation construction for $q$-ary 1-perfect codes is presented that allows us to construct $q$-ary 1-perfect codes of length $(q - 1)nm + n + m$ from the given $q$-ary 1-perfect codes of length $n =(q^{s_1} - 1) / (q - 1)$ and $m = (q^{s_2} - 1) / (q - 1)$, where $s_1, s_2$ are natural numbers not less than two. This construction allows us to also construct $q$-ary codes with parameters $(q^{s_1 + s_2}, q^{q^{s_1 + s_2} - (s_1 + s_2) - 1}, 3)_q$ and can be regarded as a $q$-ary analogue of the well-known Phelps construction.