An Algebraic Generalization of the Ramanujan Sum

TL;DR

This paper introduces an algebraic generalization of Ramanujan sums using polynomial remaindering, explores their properties, and applies them to coding theory and combinatorial problems, providing explicit formulas for specific code sizes.

Contribution

It presents a novel algebraic generalization of Ramanujan sums via polynomial remaindering and applies this to derive explicit formulas for code sizes and combinatorial sums.

Findings

Properties of the generalized Ramanujan sums are characterized.

Explicit formulas for the size of Levenshtein codes with parity conditions are provided.

Connections to applications in coding theory, DNA data storage, and the Little-Offord problem are established.

Abstract

Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial remaindering. This generalization is motivated by its applications in Restricted Partition Theory and Coding Theory. Our investigation focuses on the properties of these sums and expresses them as finite trigonometric sums subject to a coprime condition. Interestingly, these finite trigonometric sums with a coprime condition, which arise naturally in our context, were recently introduced as an analogue of Ramanujan sums by Berndt, Kim, and Zahaescu. Furthermore, we provide an explicit formula for the size of Levenshtein codes with an additional parity condition (also known as Shifted Varshamov-Tenengolts deletion correction codes), which have found many interesting applications in studying the Little-Offord problem, DNA-based data storage and distributed synchronization. Specifically, we present an explicit formula for a particularly important open case $\text{SVT}_{t,b}(s \pm \delta, 2s + 1)$ for $s$ or $s+1$ are divisible by $4$ and for small values of $\delta$.