On the error-correcting capabilities of cycle codes of graphs

The purpose is to study the error-correcting potential of cycle codes of graphs for the binary symmetric channel. Recall that the cycle code C(G) of a graph G is the binary linear block code of F₂ ^N generated by the characteristic vectors of the cycles of G, where every one of the N edges is identified with a vector coordinate. Even though one of the first things to be uncovered about such codes is that they are not optimal for growing N (for instance because, for a given rate, their minimum distance cannot grow faster than a logarithm of N), their simple structural appeal has attracted extensive study in the early days of coding theory. Today, one of the remaining open questions about cycle codes of graphs is: “for a given information rate R, what is the highest channel error probability that these codes can sustain, while achieving vanishing (with block length N) residual error probability after decoding?” The authors give a precise answer to this question