今井潤 (NTT CS Labo.)
Large-Scale Nonlinear Codes Based on Groups Derived from Permutation Puzzles and Groups Acting on Finite Designs and Their Decoding Algorithm
This paper provides a new class of binary nonlinear codes having large number of code words, as well as reasonable code length and practical minimum distances. They are realized as a kind of constant weight code derived from permutation groups whose generators are provided by permutations over nite designs as well as some nite groups acting on a nite number of symbols that correspond to the movement of a well-known permutation puzzle. This puzzle's shape is a solid body that is congruent with one of the regular polyhedra that are also known as Platonic solids. Moreover, their generalization to the other groups and its applications are provided and discussed by introducing the notion of G-spaces. A fundamental policy for constructing high-performance nonlinear codes is also discussed in more general settings by using several examples. Futhermore, their decoding algorithm is established.