Abstract
Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance ρ. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of ρ. 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 (IEEE Trans Inf Theory 40(4):1140–1146, 1994). At very high rate they are above 98% of the Shannon bound.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Solé, P., Belfiore, JC. Constructive spherical codes near the Shannon bound. Des. Codes Cryptogr. 66, 17–26 (2013). https://doi.org/10.1007/s10623-012-9633-2
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DOI: https://doi.org/10.1007/s10623-012-9633-2