definitionnovel

Generic binary subspaces of F^3

Defines generic subspaces of F3F^3: no proportional pairs, few aligned zz
non-degeneracy condition fixed by the genericity lemma
Definition·§3.2 Generic spaces of characters

Statement (paper)

A binary subspace S≤F3S\le F^3 of dimension r≥1r\ge1 is generic if no two distinct nonzero elements of SS are proportional over FF, and if, for every binary linear map g:S→F3g:S\to F^3, at most 3r3r nonzero z∈Sz\in S satisfy z⋅g(z)=z1z2z3. z\cdot g(z)=\sqrt{z_1z_2z_3}.

In plain English

A binary subspace S≤F3S\le F^3 of dimension r≥1r\ge1 is generic if no two distinct nonzero elements of SS are proportional over FF, and for every binary linear map g:S→F3g:S\to F^3 at most 3r3r nonzero z∈Sz\in S satisfy z⋅g(z)=z1z2z3z\cdot g(z)=\sqrt{z_1z_2z_3}.

Why it's here

This definition names the non-degeneracy conditions on a low-dimensional binary subspace that the later arguments need: no proportional pairs, and few solutions to the alignment equation for any linear map gg. Lemma 3.6 shows that a random subspace satisfies these conditions with high probability, so generic subspaces exist for large dd.
Classified novel:
The notion of a generic binary subspace of F^3, defined by having no proportional pairs and few solutions to the latent-alignment equation, is specific to this paper's construction. No standard definition from the literature covers this alignment condition.

Uses

Nothing else in the paper

Used by

lemRandom binary subspaces are generic

Sources

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Section
§3.2 Generic spaces of characters
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