Tree Codes and a Conjecture on Exponential Sums

TL;DR

This paper introduces a new conjecture on exponential sums and, assuming its validity, achieves the first effective construction of asymptotically good tree codes, supported by numerical evidence.

Contribution

The paper proposes a novel conjecture on exponential sums and demonstrates how it enables the construction of asymptotically good tree codes.

Findings

Numerical evidence supports the conjecture.

Codes constructed are suitable for significant-length communications.

First effective construction of asymptotically good tree codes.

Abstract

We propose a new conjecture on some exponential sums. These particular sums have not apparently been considered in the literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications.