Anirudh Krishna

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, the metric balls spanned by constant-weight vectors grow exponentially slower than those in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ \{0,1\}^{n}$ </tex-math></inline-formula>. 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.

Classically, Iceland & Samorodnitsky (2015) do just this. They generalize the beautiful Boolean analysis approach to LP bounds. Result is a bound on max rate R with relative distance δ when the parity check matrix is spanned by vectors of weight at most w. Not sure about use for small block lengths