Which codes have cycle-free Tanner graphs?

If a linear (n,k,d) code C has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n²). However, we show that cycle-free Tanner graphs cannot support good codes: we prove that d⩽max{2,2/R} for any linear code of rate R that can be represented by a cycle-free Tanner graph. Moreover, we explicitly construct codes that meet this bound with equality. We also show that all binary codes that have cycle-free Tanner graphs belong to the class of graph-theoretic cut-set codes. We conclude that the number of cycles in a Tanner graph must increase exponentially with the length n for asymptotically good codes