Dual Distance

The smallest Hamming weight of a non-zero vector in a code matrix’s kernel: the quantity every concrete LPN-variant parameter set is priced against, via the linear test framework.

Status	Problem	Tags
	EA-Code Distance over Rings Theorem 3.10 of the source proves the distance bound over \(\mathbb{F}_2\) only. Over a ring of size \(q\) its proof technique forces the rate to satisfy \(R\) much less than \(1/\ln q\), which the source states it believes is an artefact of the argument, supported by small-parameter weight-distribution experiments over \(\mathbb{F}_{17}\) and \(\mathbb{Z}_{16}\). 5 open	Code Based Cryptography Dual Distance Learning Parity With Noise Pseudorandom Correlation Functionslower-boundadaptation (ai)
	Quasi-Abelian GV, Growing Group Named as a conjecture by the source, which says the development is out of reach of the article. It is what makes the choice of group free in the QA-SD line: resistance to every linear test, for any abelian \(G\), reduces to it. Partially advanced in 2026 by concrete non-asymptotic bounds for particular groups, which restate the general question as open. 5 open	Code Based Cryptography Dual Distance Pseudorandom Correlation Functions Syndrome Decodinglower-boundadaptation (ai)
	Binary dual-distance bound is the worst case Open, and asserted rather than asked – the source writes that it expects the inequality, gives an intuition covering one of the two competing effects, records that all prior work assumes it, and proves nothing. A probe at small parameters is consistent with it; the parameters the source’s own construction uses are out of computational reach, which is why the source does not check it either. 7 open	Average Case Hardness Dual Distance Learning Parity With Noiselower-boundadaptation (ai)
	Permuted Codes Conjecture A new hardness assumption, printed as Conjecture 3.1 of the source and stated in its most general form deliberately, as a target for cryptanalysis rather than as the minimum its applications need. Implied by the older permuted puzzles conjecture; statistically true for \(O(\log n)\) samples over constant-size alphabets; false if any one of its three randomizations is dropped. 4 open	Code Based Cryptography Dual Distance Pseudorandom Codes Watermarkingassumptionbarrier (ai)