Idempotents, Mattson-Solomon polynomials and binary LDPC codes

It is shown how to construct an algorithm to search for binary idempotents that may be used to construct binary LDPC codes. The algorithm, which allows control of the key properties of sparseness, code rate and minimum distance, is constructed in the Mattson-Solomon domain. Examples are given of the codes constructed that include equivalent codes to the Euclidean and Projective Geometry codes in addition to some new codes. Codes having cycles of length 4 can also be constructed and are demonstrated to have good performance under iterative decoding.