Abstract: Encryption is carried out based on a non field, non group algebraic structure. Preferably the algebraic structure is at least one of non-associative or non-commutative. An embodiment is described in which the algebraic structure is a crossed inverse quasigroup. A crossed inverse quasigroup can be a very large quasigroup e.g. of size 1010. Either the quasigroup itself, or rules for calculating the values in the quasigroup can be distributed.

Abstract: A system and method are disclosed for modulation encoding data for storage or transmission on a multilevel medium. The method includes encoding a first portion of data using a first tier modulation code. The first tier modulation code maps a first portion of the data onto a first set of symbols. A second portion of data is encoded using a second tier modulation code. The second tier modulation maps the second portion of the data onto a second set of symbols. The second tier modulation code has error correcting characteristics. A third set of symbols is determined based on the first set of symbols and the second set of symbols. The third set of symbols is suitable to determine nominal read signal levels from a multilevel medium. Improved error characteristics are realized for encoding data for storage or transmission on a multilevel medium.

Abstract: A system and method are disclosed for modulation encoding data for storage or transmission on a multilevel medium. The method includes encoding a first portion of data using a first tier modulation code. The first tier modulation code maps a first portion of the data onto a first set of symbols. A second portion of data is encoded using a second tier modulation code. The second tier modulation maps the second portion of the data onto a second set of symbols. The second tier modulation code has error correcting characteristics. A third set of symbols is determined based on the first set of symbols and the second set of symbols. The third set of symbols is suitable to determine nominal read signal levels from a multilevel medium. Improved error characteristics are realized for encoding data for storage or transmission on a multilevel medium.

Abstract: Estimates of the horizontal and vertical positions of digital bits on film are corrected by tracking the bit transition boundaries in the horizontal and vertical directions. Frequency of useful film bit transitions in the vertical direction is maximized prior to recording by error correction encoding the digital audio bytes using a symmetrical generator polynomial and then selectively inverting symmetric portions of each codeword. Frequency of useful film bit transitions in the horizontal direction is maximized prior to recording by channel encoding the error correction symbols in each codeword so as to force the weight of the resulting symbol to be half the number of data bits per symbol.

Abstract: Estimates of the horizontal and vertical positions of digital bits on film are corrected by tracking the bit transition boundaries in the horizontal and vertical directions. Frequency of useful film bit transitions in the vertical direction is maximized prior to recording by error correction encoding the digital audio bytes using a symmetrical generator polynomial and then selectively inverting symmetric portions of each codeword. Frequency of useful film bit transistions in the horizontal direction is maximized prior to recording by channel encoding the error correction symbols in each codeword so as to force the weight of the resulting symbol to be half the number of data bits per symbol.

Abstract: Error correction for polynomial block codes is achieved without prior evaluation of power sum symmetric functions. The received word R (z) is reduced mod G (z), the generator of the code and a function F (z) of error locator polynomial W(z), errata values Y and code dependent functions f(x.sub.i) of the error positions x.sub.i given by ##EQU1## is decomposed into a rational polynomial function N (z)/W (z) for which deg (N (z) )<deg ( W (z) )<number of correctable errors. W (z) is the error locator polynomial, the roots of which are the errata locations X and Y, the correction to the received character is obtained from ##EQU2## evaluated at X.sub.i using non-erased check symbols of R (z). Correction is carried out in a crossbar switch structure which recalls a stored copy of R (z) and corrects bits as specified by (X.sub.i, Y.sub.i).