  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 (2006), we use this fact to
improve on the first linear programming bound on the rate of LDPC codes, as the
function of their minimal distance. This improvement, combined with the
techniques of Ben-Haim and Lytsin (2006), improves the rate vs distance bounds
for LDPC codes in a significant sub-range of relative distances.
