Constructive spherical codes near the Shannon bound

TL;DR

This paper presents a new polynomial-time construction of spherical codes that nearly reach Shannon's bound, improving previous results and combining geometric and algebraic coding techniques.

Contribution

It introduces a novel construction method for spherical codes that are close to the Shannon bound, outperforming earlier approaches.

Findings

Codes are polynomial time constructible.

Achieve over 98% of Shannon bound at high rates.

Parameters outperform previous constructions by Lachaud and Stern.

Abstract

Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance $\rho$. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of $\rho$. The construction is based on the Yaglom map combined with some finite sphere packings obtained from nonconstructive codes for the Euclidean metric. Concatenating geometric codes meeting the TVZ bound with a Lee metric BCH code over $GF(p),$ we obtain spherical codes that are polynomial time constructible. Their parameters outperform those obtained by Lachaud and Stern in 1994. At very high rate they are above 98 per cent of the Shannon bound.