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In this post, I asked a question regarding solutions to system of binary equations. The answer that was accepted provided a complete structural solution to the question. I have been working on a derivative of this problem that follows.

Given a matrix $A$ as defined in the initial problem. Is there an algebraic description of all $z$ values for which the algorithm described in the accepted answer yields a non empty solution space?

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Joseph Zambrano
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  • Calculation is done in $\mathbb{Z}_2$? – Markus Scheuer Sep 02 '16 at 07:00
  • No, ${0,1}\in\mathbb{Q}$ or $\mathbb{R}$. Although it seems there's often a clean relation between the two. – Joseph Zambrano Sep 02 '16 at 11:41
  • Ok, I see. Thanks. – Markus Scheuer Sep 02 '16 at 11:54
  • @MarkusScheuer If you have any insight on these please let me know. I will likely award the bounty as there has not been much interest. – Joseph Zambrano Sep 03 '16 at 14:56
  • @Joseph Zambrano : if $x \cdot y \ne 0$ then $(x-y)\cdot \mathrm{row}i(A) \ne \sum{j=1}^n x_j$ , since at least one of the $x_j = 1$ will be canceled on the LHS. i.e. If $z_i = 1$ then $x$ and $y$ must be normal. –  Sep 04 '16 at 10:43
  • Yes I agree. In thing I can add is that the set of all possible $z$ values is parameterized by $s\in{-1,0,1}^n$ in the following way: $z_i(s)=\prod_k (1-1/2(s_k^2+s_k)+s_ka_{i,k})$ for each $i\in [m]$. However, it is often the case that $z_i(s)=z_i(s')$ for two distinct $s$ and $s'$. – Joseph Zambrano Sep 04 '16 at 13:32

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