On Coset Leader Graphs of LDPC Codes

Our main technical result is that, in the coset leader graph of a linear binary code of block length n, the metric balls spanned by constant-weight vectors grow exponentially slower than those in {0, 1}(n). Following the approach of Friedman and Tillich, we use this fact to improve on the first linear programming bound on the rate of low-density parity check (LDPC) codes, as the function of their minimal relative distance. This improvement, combined with the techniques of Ben-Haim and Litsyn, improves the rate versus distance bounds for LDPC codes in a significant subrange of relative distances.