Algebraic and Combinatorial Methods for Reducing the Number of Variables of Partially Defined Discrete Functions

Applications of pattern recognition, design of faulttolerant systems and communications have key problems that arenaturally described by partially defined (incompletely defined)discrete functions. Such partially defined functions arising frompractical demands usually have a large number of variables andso their direct implementations require complex systems. Thusit is important to have at hand an efficient method to reducethe number of their variables. Here we review recent results tolinearly decompose a discrete function using a transform thatcan be efficiently implemented as a Galois field deconvolution. We also study the question: What are the general bounds for thedimension of the range space for an arbitrary linear transformto reduce a partially defined discrete function? We derive abound for the dimension of the range for arbitrary lineartransformation. We also estimate how good linear decompositioncan be obtained by the use of random transformations and showthat with a randomly generated transform we can reach theabove discussed bound.