Anirudh Krishna
@anirudhkrishna.bsky.social
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
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.
ieeexplore.ieee.org
April 4, 2025 at 7:36 PM
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