On Binary Shadow Codes

TL;DR

This paper generalizes shadow codes to include higher-degree polynomials and demonstrates that simpler polynomials can improve minimum distance bounds, but overall shadow codes are less efficient than Reed-Solomon and Reed-Muller concatenations.

Contribution

It extends shadow code theory to arbitrary polynomial degrees and compares their performance to classical concatenated codes.

Findings

Higher-degree polynomials do not significantly improve code performance.

Restricting to degree one or less yields better minimum distance bounds.

Shadow codes are less efficient than Reed-Solomon and Reed-Muller concatenations.

Abstract

We generalize the shadow codes of Cherubini and Micheli to include basic polynomials having arbitrary degree, and show that restricting basic polynomials to have degree one or less can result in improved lower bounds on the minimum distance of the code. However, even these improved lower bounds suggest that shadow codes have considerably inferior distance-rate characteristics compared with the concatenation of a Reed-Solomon outer code and a first-order Reed-Muller inner code.