Encoding Spatially Coupled LDPC codes with Polynomial Generator Matrices

Spatially coupled low-density parity-check (SC-LDPC) codes have recently attracted a lot of attention because of their optimal asymptotic performance. Time-invariant and periodically time-varying SC-LDPC codes are particularly interesting code families, since they have good finite-length performance and allow for a very compact representation. However, one of the key challenges to make SC-LDPC codes practical consists in finding efficient techniques for their encoding. In this paper, we propose a method to construct polynomial generator matrices for time-invariant and periodically time-varying SC-LDPC codes, based on their polynomial parity-check matrix. We show that it is always possible to find a polynomial generator matrix in quasi-standard form, where the systematic part consists of a diagonal matrix with identical entries, enabling efficient and non-catastrophic encoding. We also show that it is always possible to compute an equivalent rational function generator matrix in standard form that guarantees recursive, systematic (and therefore also non-catastrophic) encoding. Finally, to illustrate the method, we introduce a family of time-invariant SC-LDPC codes characterized by a binary parity-check matrix associated to a Tanner graph with girth larger than 4 and by good asymptotic and finite length performance. These codes naturally admit a quasi-standard polynomial generator matrix that, depending on the number of control symbols per period, either can be efficiently converted into polynomial standard form or remains in quasi-standard form but is relatively sparse. Both these features enable fast and efficient encoding.