Which Codes Have Cycle-free

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 2). However, we show that cycle-free Tanner graphs cannot support good codes. Specically, 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 d n k + 1 + n + 1 k + 1 < 2 R Furthermore, we show by means of an explicit construction that this bound is tight for all values of n and k. We also prove that binary codes which have cycle-free Tanner graphs belong to the class of graph-theoretic codes, known as cut-set codes of a graph. Finally, we discuss the asymptotics for Tanner graphs with cycles, and present a number of open problems for future research.