Cyclonomial Number Systems and the Ranking of Lexicographically Ordered Constant-Sum Codes

Cyclonomial coefficients are defined as a generalization of binomial coefficients. It is proved that each natural number can be expressed, in a unique way, as the sum of cyclonomial coefficients, satisfying certain conditions. This cyclonomial number system generalizes the well-known binomial number system. It appears that this system is the appropriate number system to index the words of the lexicographically ordered code \(L^q(n, k)\). This code consists of all words of length \(n\) over an alphabet of \(q\) symbols, such that the sum of the digits is constant. It provides efficient algorithms for the conversion of such a codeword to its index, and vice versa.