Stopping sets and the girth of tanner graphs

Recent work has related the error probability of iterative decoding over erasure channels to the presence of stopping sets in the Tanner graph of the code used. In particular, it was shown that the smallest number of uncorrected erasures is the size of the graph's smallest stopping set. Relating stopping sets and girths, we consider the size σ(d, g) of the smallest stopping set in any bipartite graph of girth g and left degree d. For g ≥ 8 and any d, we determine σ(d, g) exactly. For larger gs we bound σ(d, g) in terms of d, showing that for fixed d, σ(d, g) grows exponentially with g. Since constructions of high-girth graphs are known, one can therefore design codes with good erasure-correction guarantees under iterative decoding.