Majority Logic Coding and its Multinomial Representation

Multinomial representations are derived for majority logic operations on bipolar binary data. The coefficients are given simply in terms of the readily computed lower Cholesky factor of Pascal Matrices of order n for codes of block length n. Sufficient conditions on majority logic codes having no deterministic errors introduced in the majority logic coding and majority logic decoding process are given in terms of the multinomial coefficients and codewords. These are used to guarantee that certain majority logic codes based on Hadamard matrices, Rademacher matrices and pseudo-random bipolar sequences have no deterministic coding/decoding errors. Building on this work, a novel majority logic coding scheme is proposed which comprises a first stage of linear coding with its own spreading factor, followed by a second stage of majority logic coding with its own spreading factor, and a majority logic based decoding to recover the data.