Unweighted linear congruences with distinct coordinates and the Varshamov--Tenengolts codes

TL;DR

This paper provides explicit formulas for solutions of unweighted linear congruences with distinct coordinates and applies these results to derive formulas for the number of codewords in Varshamov--Tenengolts codes, enhancing understanding of their combinatorial properties.

Contribution

It introduces explicit formulas for solutions of certain linear congruences and applies them to analyze the structure and enumeration of Varshamov--Tenengolts codes.

Findings

Explicit formulas for solutions of unweighted linear congruences with distinct coordinates

Derived formulas for the number of codewords with fixed Hamming weight in VT codes

Connected combinatorial problems to the properties of these codes

Abstract

In this paper, we first give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov--Tenengolts code $VT_b(n)$ with Hamming weight $k$, that is, with exactly $k$ $1$'s. The Varshamov--Tenengolts codes are an important class of codes that are capable of correcting asymmetric errors on a $Z$-channel. As another application, we derive Ginzburg's formula for the number of codewords in $VT_b(n)$, that is, $|VT_b(n)|$. We even go further and discuss connections to several other combinatorial problems, some of which have appeared in seemingly unrelated contexts. This provides a general framework and gives new insight into all these problems which might lead to further work.