Random Research HighlightHyperdense coding. The modulo 6 representation of polynomial f is just any polynomial f+6g, that is, we do not care about terms multiplied by 6. The 6-strong representation of f modulo 6 is polynomial f+2g+3h, where no two of g, f and h have common monomials. Some surprising applications are given: it is shown that n homogeneous linear polynomials can be linearly transformed to o(n) linear polynomials, such that from these linear polynomials one can get back the 6-strong representations of the original ones with linear transformations. Probabilistic Memory Cells (PMC's) are defined, and it is shown that one can encode n bits into n PMC's, transform n PMC's to o(n) PMC's ( we call this Hyperdense Coding), and one can transform back these o(n) PMC's to n PMC's, and from these one can get back the original bits.  A method is given for converting n x n matrices to o(n) x o(n) matrices  and from these tiny matrices one can retrieve 6-strong representations of the original ones with linear transformations. Note: This is not an LZW-like data compression method: We can compress and reconstruct Kolmogorov-random sequences as well.
IEEE Transactions on Information Theory.Volume 54, Issue 8, pp. 3687-3692, Aug. 2008