On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes

This paper deals with the design of low-rate sparse-graph codes with linear minimum distance (dmin) in the blocklength. First, we define a necessary condition that a quite general family of graphical codes has to satisfy in order to have linear dmin.The condition generalizes results known for turbo codes [9] and LDPC codes. Secondly, we try to justify the necessity to introduce degree-1 bits (transmitted or punctured) into the code structure, while designing an efficient low-rate code. As a final result of our investigation, we present a new ensemble of low-rate codes, designed under the necessary condition and having bits of degree 1. The asymptotic analysis of the ensemble shows that its iterative threshold is close to the Shannon limit. It also has linear dmin, a simple structure and enjoys a low decoding complexity and a fast convergence.