On the subset sum problem over finite fields

TL;DR

This paper investigates the subset sum problem over finite fields, providing explicit and asymptotic formulas for solution counts, with implications for decoding Reed-Solomon codes.

Contribution

It offers new mathematical formulas for counting solutions to the subset sum problem over finite fields, enhancing understanding of its structure and decoding applications.

Findings

Derived explicit formulas for solution counts in specific cases

Established asymptotic estimates for the number of solutions

Connected solution counts to Reed-Solomon code decoding

Abstract

The subset sum problem over finite fields is a well-known {\bf NP}-complete problem. It arises naturally from decoding generalized Reed-Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematical point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we obtain some results on the decoding problem of Reed-Solomon codes.