On Some Classes of Functions and Hypercubes

X iv :0 81 2. 43 67 v1 [ cs .D M ] 2 3 D ec 2 00 8 ON SOME CLASSES OF FUNCTIONS AND HYPERCUBES DIMITER STOICHKOV KOVACHEV Abstra t. In this paper, some lasses of dis rete fun tions of k-valued logi are onsidered, that depend on sets of their variables in a parti ular way. Obtained results allow to " onstru t" these fun tions and to present them in their tabular, analyti al or matrix form, that is, as hyper ubes, and in parti ular Latin hyper ubes. Results onne ted with identifying of variables of some lasses of fun tions are obtained. 1. Introdu tion, Definitions and Notation Let Ek = {0, 1, . . . , k− 1}, k ≥ 2. The set of all fun tions of n variables of k−valued logi is denoted by P k n , where P k n = {f : E k −→ Ek}. A matrix of m rows and m olumns is denoted by ||aij ||1 and is alled a 2−dimensional matrix of order m. By ||bi1i2...in||1 we will denote the n− dimensional matrix of order k, whi h is referred to as an n−dimensional hyper ube of order k by some authors [5℄. Ea h fun tion f(x1, x2, . . . , xn) ∈ P k n , by using the equality (1) ai1i2...in = f(x1 = i1 − 1, x2 = i2 − 1, . . . , xn = in − 1), an be presented in the matrix form ||ai1i2...in ||1 as an n− dimensional hyper ube of order k, based on the set Ek. Latin squares and hyper ubes have their appli ations [5℄ in oding theory, error orre ting odes, information se urity, de ision making, statisti s, ryptography, on i t-free a ess to parallel memory systems, experiment planning, tournament design, et . Ea h n−dimensional matrix A = ||ai1i2...in||1 of order k is alled a Latin (Permutational) n−dimensional hyper ube of order k, based on the set Ek, if for ea h s, s = 1, 2, . . . , n, we have ∣∣∣ ∣ ∣ k ⋃