Which codes have cycle-free Tanner graphs?

If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n). However, we show that cycle-free Tanner graphs cannot support good codes. Specifically, let C be an (n, k, d) linear code of rate R = k/n that can be represented by a Tanner graph without cycles. We prove that if R ≥ 0.5 then d ≤ 2, while if R < 0.5 then C is obtained from a code of rate ≥ 0.5 and distance ≤ 2 by simply repeating certain symbols. In the latter case, we prove that