Modular Representations of Polynomials: Hyperdense Coding and Fast Matrix Multiplication

Vince Grolmusz
2008 IEEE Transactions on Information Theory  
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 n o(1) linear polynomials, such that from these linear
more » ... ynomials 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 n o(1) PMC's ( we call this Hyperdense Coding), and one can transform back these n o(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 n o(1) bits. A method is given for converting n × n matrices to n o(1) × n o(1) matrices and from these tiny matrices one can retrieve 1-astrong representations of the original ones, also with linear transformations. Applying PMC's to this case will return the original matrix, and not only the representation.
doi:10.1109/tit.2008.926346 fatcat:qvvq7tkz7zhgzederxpkkiopae