Modular Representations of Polynomials: Hyperdense Coding and Fast Matrix Multiplication

A certain modular representation of multilinear polynomials is considered. The modulo 6 representation of polynomial f is just any polynomial f + 6g. The 1-a-strong representation of f modulo 6 is polynomial f + 2g + 3h, 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.