Pearson codes

TL;DR

This paper analyzes the properties and constructions of optimal Pearson codes, which are specialized $q$-ary code sets used with Pearson distance for error correction in noisy channels with unknown gain and offset.

Contribution

It provides a detailed analysis of optimal Pearson codes, compares their redundancy with T-constrained codes, and identifies conditions under which these codes are optimal.

Findings

For q ≤ 3, 2-constrained codes are optimal Pearson codes.

For q ≥ 4, 2-constrained codes are not optimal.

Optimal Pearson codes have lower redundancy than prior T-constrained codes in certain cases.

Abstract

The Pearson distance has been advocated for improving the error performance of noisy channels with unknown gain and offset. The Pearson distance can only fruitfully be used for sets of $q$-ary codewords, called Pearson codes, that satisfy specific properties. We will analyze constructions and properties of optimal Pearson codes. We will compare the redundancy of optimal Pearson codes with the redundancy of prior art $T$-constrained codes, which consist of $q$-ary sequences in which $T$ pre-determined reference symbols appear at least once. In particular, it will be shown that for $q\le 3$ the $2$-constrained codes are optimal Pearson codes, while for $q\ge 4$ these codes are not optimal.