Hardness of Approximation Results for the Problem of Finding the Stopping Distance in Tanner Graphs

Abstract

Tanner Graph representation of linear block codes is widely used by iterative decoding algorithms for recovering data transmitted across a noisy communication channel from errors and erasures introduced by the channel. The stopping distance of a Tanner graph T for a binary linear block code C determines the number of erasures correctable using iterative decoding on the Tanner graph T when data is transmitted across a binary erasure channel using the code C. We show that the problem of finding the stopping distance of a Tanner graph is hard to approximate within any positive constant approximation ratio in polynomial time unless P=NP. It is also shown as a consequence that there can be no approximation algorithm for the problem achieving an approximation ratio of $2^{(\log n)^{1-\epsilon}}$ for any ε> 0 unless NP ⊆ DTIME(n ^poly(logn)).