Minimum-distance upper bound for binomial permutation-polynomial LDPC codes

Investigate the minimum-distance upper bound for protograph-based LDPC codes whose parity-check matrices use noncommuting binomial permutation polynomials of the form $f(x)=ax^{(q+1)/2}+bx$ over $mathbb{F}_q$, determining whether the upper bound that applies to QC-LDPC codes is replaced by a different bound or does not apply.

Background

The paper constructs protograph-based LDPC codes using the family of binomial permutation polynomials $f(x)=ax^{(q+1)/2}+bx$ over an odd finite field. Unlike monomial and affine constructions, these permutation polynomials do not generally commute under composition, so their associated permutation matrices also do not commute.

For monomial-based constructions and QC-LDPC codes, commutativity yields a known upper bound on minimum distance. The authors note that noncommutativity removes the direct applicability of this QC-LDPC upper bound, but they do not establish what minimum-distance behavior or replacement bound holds for the binomial construction. This unresolved issue is identified as a subject for future investigation.