Fast systematic encoding of quasi-cyclic codes using the Chinese remainder theorem

Quasi-cyclic (QC) codes are a wide class of error-correcting codes possessing nice theoretical properties and having many practical applications. This paper provides a new approach to the problem of efficient encoding of QC codes based on the Chinese remainder theorem (CRT). We present a fast systematic CRT-based encoding algorithm that has superior asymptotic complexity than the previous methods based on shift registers. We also consider the encoding problem for QC low-density parity-check (LDPC) codes. In the special case when the parity part of a sparse parity-check QC matrix has a QC generalized inverse we propose a systematic CRT-based encoding algorithm that can exploit the parity-check matrix sparseness. We also give necessary and sufficient conditions when a QC matrix over an arbitrary field has a QC generalized inverse of the same circulant size.