An Algebraic Approach to Reducing the Number of Variables of Incompletely Defined Discrete Functions
Jaakko T. Astola, Pekka Astola, Radomir S. Stanković, Ioan Tăbuş · 2016
In this paper, we consider incompletely defined discrete functions, i.e., Boolean and multiple-valued functions, f: S→{0,1,,q -- 1} where S ⊆ {0,1,,q -- 1}n i.e., the function value is specified only on a certain subset S of the domain of the corresponding completely defined function. We assume the function to be sparse i.e. |S| is 'small' relative to the cardinality of the domain. We show that by embedding the domain {0,1,,q -- 1}n, where n is the number of variables and q is a prime power, in a suitable ring structure, the multiplicative structure of the ring can be used to construct a linear function {0,1,,q -- 1}n→{0,1,,q -- 1}m that is injective on S provided that m > 2logq|S|+logq(n -- 1). In this way we find a linear transform that reduces the number of variables from n to m, and can be used e.g. in implementation of an incompletely defined discrete function by using linear decomposition.