Asymptotic bounds for spherical codes

Yu. I. Manin ; M. Marcolli

Izvestiya. Mathematics , Tome 83 (2019) no. 3, pp. 540-564

Résumé

The set of all error-correcting codes $C$ over a fixed finite alphabet $\mathbf{F}$ of cardinality $q$ determines the set of code points in the unit square $[0,1]^2$ with coordinates $(R(C), \delta (C))$:= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of “good codes” and comparing new classes of codes with earlier ones. A less classical approach studies the geometry of the whole set of code points $(R,\delta)$ (with $q$ fixed), at first independently of its computability properties, and only afterwards turning to problems of computability, analogies with statistical physics, and so on. The main purpose of this article consists in extending this latter strategy to the domain of spherical codes.

Détail

Keywords: error-correcting codes, spherical codes, asymptotic bounds.