Designing a Good Low-Rate Sparse-Graph Code

This paper deals with the design of low-rate sparse-graph codes, having a linear minimum distance d_{min} in the blocklength n. Its main contributions are: a) a necessary condition on a general family of sparse-graph codes with linear d_{min}; b) a justification of having degree-1 bits in the low-rate code structure; c) a new, efficient ensemble of low-rate sparse-graph codes with bits of degree 1, designed so that the necessary condition (a) is satisfied.