Message passing soft decoding of linear block codes over arbitrary finite fields

In this paper, soft-input/soft-output decoding of an arbitrary linear block error correction code over any finite field is performed by transforming an algebraic description of the decoding problem to a binary message passing decoding problem, such as used for a binary LDPC decoder. Field symbols are represented using a Hadamard-like transform of the probability distribution over the field, and field operations in this setting are represented by real multiplication and permutations. This gives rise to a “soft key equation,” an overdetermined nonlinear equation. Here, a solution to the soft-key equation is obtained by enforcing a parity consistency, which converts the decoding problem into a message-passing problem which is able to exploit the sparse structure of the problem representation. Simulated performance results indicate that decoding is strongest for highest rate codes, motivating the investigation of improved performance.