claimroutine

Alignment reduces to column-span membership

Alignment on

E

forces

c

into the column span of

M

reduces alignment to a linear-algebra condition

Claim·§3.2 Generic spaces of characters·inside lemRandom binary subspaces are generic

Statement (paper)

Let

E\subset\F_2^r

be a set of more than

3r

distinct nonzero inputs, and take an injective parametrization

z_i(t)=\sum_j x_{ij}t_j

. If some binary linear map

g

satisfies the alignment condition on all of

E

, then the vector

c=\bigl(\sqrt{z_1(t)z_2(t)z_3(t)}\bigr)_{t\in E}

lies in the

F

-column span of the

3r

-column matrix

M_{t,(i,j)}=t_jz_i(t)

t\in E

1\le i\le 3

1\le j\le r

. Conversely, membership of

c

in this column span supplies the basis values of such a

g

, so this column-membership condition accounts for all possible maps

g

In plain English

For a set

E\subset\F_2^r

of more than

3r

distinct nonzero inputs, if some binary linear map

g

satisfies the alignment condition on all of

E

, then the vector

c=(\sqrt{z_1(t)z_2(t)z_3(t)})_{t\in E}

lies in the

F

-column span of the

3r

-column matrix

M_{t,(i,j)}=t_jz_i(t)

. Conversely, membership of

c

in that span yields the basis values of such a

g

, so this condition covers every possible

g

Why it's here

This turns the question of whether any aligned linear map exists into a single linear-algebra condition on the

x_{ij}

, namely whether

c

lies in the column span of

M

. That condition can then be handled by treating the

x_{ij}

as indeterminates, as the next claim does with non-membership over the function field, to show a random subspace is generic with high probability.}</br>

Classified routine:

This step rewrites the alignment condition, linear in the unknown basis values of g, as a column-span membership statement for an explicit matrix. It is a direct linear-algebra reformulation inside the genericity proof and uses no new idea.

Uses

Nothing else in the paper

Used by

claimGeneric non-membership over function field

Sources

PaperOpen

Section: §3.2 Generic spaces of characters
Page: —
Label: —
Line: sections/02-latent.tex:277

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