Constructions of A Large Class of Optimum Constant Weight Codes over F_2

TL;DR

This paper introduces a new generalized construction method for optimum constant weight codes over F_2, achieving bounds and presenting large classes of such codes for various parameters, with conjectures on their optimality.

Contribution

It presents a novel generalized $(u, u+v)$ construction method for constant weight codes over F_2, meeting known bounds and proposing conjectures on their optimality for various code parameters.

Findings

Constructed codes meet Brouwer and Verhoeff bound.

Presented classes of optimal codes for k=2 to 6, n ≤ 128.

Proposed conjectures on the optimality of constructed codes.

Abstract

A new method of constructing optimum constant weight codes over F_2 based on a generalized $(u, u+v)$ construction is presented. We present a new method of constructing superimposed code $C_{(s_1,s_2,\cdots,s_I)}^{(h_1, h_2, \cdots, h_I)}$ bound. and presented a large class of optimum constant weight codes over F_2 that meet the bound due to Brouwer and Verhoeff, which will be referred to as BV . We present large classes of optimum constant weight codes over F_2 for $k=2$ and $k=3$ for $n \leqq 128$. We also present optimum constant weight codes over F_2 that meet the BV bound for $k=2,3,4,5$ and 6, for $n \leqq 128$. The authors would like to present the following conjectures : $C_{I}$: $C_{(s_1)}^{(h_1)}$ presented in this paper yields the optimum constant weight codes for the code-length $n=3h_1$, number of information symbols $k=2$ and minimum distance $d=2h_1$ for any positive integer $h_1$. $C_{II}$: $C_{(s_1)}^{(h_1)}$ yields the optimum constant weight codes at $n=7h_1, k=3$ and $d=4h_1$ for any $h_1$. $C_{III}$: Code $C_{(s_1,s_2,\cdots,s_I)}^{(h_1, h_2, \cdots, h_I)}$ yields the optimum constant weight codes of length $n=2^{k+1}-2$, and minimum distance $d=2^{k}$ for any number of information symbols $k\geq 3$.