It is an overview of results on classification and enumeration problems of discrete functions by general linear and affine groups for a period from 1950 to 2005. An enumeration problem is to compute a number of equivalence classes of functions (orbits under a group action). A classification problem is to obtain a complete list of orbit representatives and to describe equivalence classes. For the first period there was obtained main enumeration formulas and a classification of four and five variables Boolean functions. But the further progress in classification results was limited by the great number of functions. Last time a main attention was concentrated on the enumeration and classification technique for cosets of Reed-Muller codes of bounded degree. Due to this approach there was constructed a classifications of cosets the Reed-Muller codes of degree three (cubic forms) of nine variables and forms of degree four of eight variables.

classification, invariant, enumeration, discrete functions, Reed-Muller codes

Mathematical problems and theory of numerical methods