Cyclotomic idempotent-based binary cyclic codes

It is shown that idempotents based on cyclotomic cosets may be used to construct some new one-step majority-logic codes. The construction method produces the dual code idempotent which can be used to directly define the parity-check matrix of the cyclic code. A feature of the cyclotomic idempotent codes is the incremental approach to the sparseness of the parity-check matrix and the property of parity-check bit orthogonality which is known to be useful in belief-propagation decoding. These codes have high minimum distance and a performance 0.4 dB away from the theoretical limit is demonstrated.