Non-binary LDPC decoding using truncated messages in the Walsh-Hadamard domain

The Extended Min-Sum (EMS) algorithm for non-binary low-density parity-check (LDPC) defined over an alphabet of size $q$ operates on truncated messages of length $q'$ to achieve a complexity of the order $q'^2$. In contrast, Walsh-Hadamard (WH) transform based iterative decoders achieve a complexity of the order $q\log q$, which is much larger for $q'<<q$. In this paper, we demonstrate that considerable savings can be achieved by letting WH based decoders operate on truncated messages as well. We concentrate on the direct WH transform and compute the number of operations required if only $q'$ of the $q$ inputs are non-zero. Our paper does not cover the inverse WH transform and hence further research is needed to construct WH based decoders that can compete with the EMS algorithm on complexity terms.

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