By Topic

Modular Representations of Polynomials: Hyperdense Coding and Fast Matrix Multiplication

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

1 Author(s)
Grolmusz, V. ; Dept. of Comput. Sci., Eotvos Univ., Budapest

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.

Published in:

Information Theory, IEEE Transactions on  (Volume:54 ,  Issue: 8 )