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 x1, x2, . . . , xn can be linearly transformed to no(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 (PMC's) are also defined here, and it is shown that one can encode n bits into n PMC's, transform n PMC's to no(1) PMC's ( we call this Hyperdense Coding), and one can transform back these no(1) PMC's to n PMC's, and from these how one can get back the original bits, while from the hyperdense form one could have got back only no(...

• Type: Journal
• Year: 2008
• Where: TIT
• Authors: Vince Grolmusz
• Keywords: 1-a-strong Representations, Linear Polynomials, Polynomials, TIT 2008