Graph-covers and iterative decoding of finite length codes

Codewords in finite covers of a Tanner graph G are characterized. Since iterative, locally operating decoding algorithms cannot distinguish the underlying graph G from any covering graph, these codewords, dubbed pseudo-codewords are directly responible for sub-optimal behavior of iterative decoding algorithms. We give a simple characterization of pseudocodewords from finite covers and show that, for the additive, white Gaussian noise channel, their impact is captured in a finite set of “minimal” pseudocodewords. We also show that any (j, k)-regular graph possesses asymptotically vanishing relative minimal pseudo-weight. This stands in sharp contrast to the observation that for j > 2 the minimum Hamming distance of a (j, k)-regular low-density parity-check code typically grows linearly with the length of the code.