# Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length

## TL;DR

This paper introduces new design methods for time-invariant SC-LDPC convolutional codes that achieve smaller constraint lengths by directly designing the syndrome former matrix, supported by theoretical bounds and practical techniques.

## Contribution

It presents a direct design approach for SC-LDPC-CCs that results in smaller constraint lengths and provides theoretical bounds and new techniques to approach these limits.

## Key findings

- New design techniques approach theoretical lower bounds.
- Codes with smaller syndrome former constraint lengths.
- Enhanced understanding of cycle constraints in Tanner graphs.

## Abstract

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check (QC-LDPC) block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.