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Let $v_1v_2...v_n$ be a sequence of vectors in $\mathbb{F}_2^{m}$. We say that this sequence is neighboring orthogonal if $\langle v_i, v_{i+1}\rangle = 0$ for each $i\in \{1,...,n\}$, where for each $u,u'\in \mathbb{F}_2^m$, $\langle u, u'\rangle = \sum_{j}u_ju_j'$.

In other words, each two consecutive vectors are orthogonal, but we don't impose orthogonality for non-adjacent vectors.

A natural problem involving neighboring sequences is the following.

Problem 1: Given subspaces $V_1,...,V_n$ of $F_2^{m}$, determine whether there exist vectors $v_1\in V_1$, ... ,$v_n\in V_n$ such that $v_1v_2...v_n$ forms a neighboring orthogonal sequence.

Questions:

  1. In which branches of mathematics neighboring sequences occur?
  2. What is the standard name neighboring sequences?
  3. Has Problem 1 been studied before?

I would also be interested in variants of the above questions when we use $\mathbb{R}^m$ in place of $\mathbb{F}_2^m$.

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    $\begingroup$ What's ${\mathbb R}_2$? $\endgroup$ – Noam D. Elkies Jul 4 '17 at 2:57
  • $\begingroup$ A typo. $\mathbb{R}_2$ used to be a typo. But now it no longer is. $\endgroup$ – verifying Jul 4 '17 at 7:42

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