Modular Representations of Polynomials: Hyperdense Coding and Fast Matrix Multiplication

Vince Grolmusz

doi:10.1109/tit.2008.926346

Abstract

A certain modular representation of multilinear polynomials is considered. The modulo 6 representation of polynomial f is just any polynomial f + 6 g . The 1-a-strong representation of f modulo 6 is polynomial f + 2 g + 3 h , where no two of g , f , and h have common monomials. Using this representation, some surprising applications are described: it is shown that n homogeneous linear polynomials x 1 , x 2 ,..., x n can be linearly transformed to n o(1) linear polynomials, such that from these linear polynomials one can get back the 1-a-strong representations of the original ones, also with linear transformations. Probabilistic Memory Cells (PMCs) are also defined here, and it is shown that one can encode n bits into n PMCs, transform n PMCs to n o(1) PMCs (we call this Hyperdense Coding), and one can transform back these n o(1) PMCs to n PMCs, and from these how one can get back the original bits, while from the hyperdense form one could have got back only n o(1) bits. A method is given for converting n times n matrices to n o(1) times n o(1) matrices and from these tiny matrices one can retrieve 1-a-strong representations of the original ones, also with linear transformations. Applying PMCs to this case will return the original matrix, and not only the representation.

Full Text