On some batch code properties of the simplex code

TL;DR

This paper explores the batch code properties of binary simplex codes, providing a constructive proof of an intermediate property and relating it to additive problems in finite abelian groups, while proposing a generalized conjecture.

Contribution

It offers a simple, constructive proof of an intermediate batch code property for simplex codes and introduces a generalized conjecture in finite abelian groups.

Findings

Established a new property of simplex codes related to batch coding

Connected batch code properties to additive problems in finite abelian groups

Proposed a conjecture generalizing existing batch code conjectures

Abstract

The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture.