Low-rate LDPC convolutional codes with short constraint length

We study a family of LDPC convolutional codes having code rate of the type 1/a, and analyze their minimum distance and local cycles length properties. We consider some low weight codewords that are known from the literature, and are easily obtained from the symbolic parity-check matrix of these codes. Starting from the structure of such codewords, we follow a twofold approach: i) we exploit graph-based techniques to design these codes with the aim to maximize their minimum distance while keeping the syndrome former constraint length as small as possible and ii) we provide a simple form for their generator matrices that allows to perform exhaustive searches through which we verify that the code design actually reaches its target. We also estimate the normalized minimum distance multiplicity for the codes we consider, and introduce the notion of symbolic graphs as a new tool to study the code properties.