Time-Invariant Spatially Coupled Low-Density Parity-Check Codes with Small Constraint Length

TL;DR

This paper introduces a direct design method for time-invariant SC-LDPC codes that reduces constraint length and establishes theoretical lower bounds, improving upon existing solutions.

Contribution

It presents a novel direct design approach for syndrome former matrices, bypassing QC code constraints, and derives lower bounds on constraint length using a new numerical representation.

Findings

Achieved smaller constraint lengths than previous best solutions.

Developed a new numerical approach for analyzing syndrome former matrices.

Established theoretical lower bounds on constraint length for these codes.

Abstract

We consider a special family of SC-LDPC codes, that is, time-invariant LDPCC codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from QC block codes, and applying suitable unwrapping procedures. We show that, by directly designing the LDPCC code syndrome former matrix without the constraints of the underlying QC block code, it is possible to achieve smaller constraint lengths with respect to the best solutions available in the literature. We also find theoretical lower bounds on the syndrome former constraint length for codes with a specified minimum length of the local cycles in their Tanner graphs. For this purpose, we exploit a new approach based on a numerical representation of the syndrome former matrix, which generalizes over a technique we already used to study a special subclass of the codes here considered.