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Suppose we know $deg(m(x))=n-1=deg(f_1(x))=deg(f_2(x))$. Suppose we know $C_1(x),C_2(x)$ where $deg(C_i)=n$. Then given $n$ values of $$C_1(x)(x+1)m(x) +C_1(x)(x+2)f_1(x)\in\Bbb F_q[x]$$ and $n$ ...

If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...