No reduction from Search noisy $k$-LIN to Noisy k-LIN

A reduction of class unstated from [[noisy-k-lin-over-expanders#search-noisy-k-lin|Search noisy $k$-LIN]] to Noisy k-LIN would imply a contradiction.

Statement

Migrated verbatim from noisy-k-lin-over-expanders § Search noisy $k$-LIN:

The search variant asks to recover $\mathbf{s}$ from $(\mathbf{M},\mathbf{Ms}+\mathbf{e})$. The search-to-decision reduction for standard LPN does not immediately transfer to the expanding-matrix setting. GHJS25 Theorem 8.8 uses a search variant as an alternative assumption sufficient for PKE under the joint conjecture with planted clique.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• “The search-to-decision reduction for standard LPN does not immediately transfer to the expanding-matrix setting” is an ABSENCE of a result, not a barrier: nothing is proved impossible. It cannot be typed in the (exists reduction of class C) ⇒ Q form.
• No citation.