This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated!

**The problem**:
Let $m\geq8$ be an even integer and $\varepsilon,\alpha\in\left( 0,1\right)
$ be two fixed constants. For each $m$ large enough, construct or an orthogonal matrix $\mathbf{T}_{m}=\left( \gamma_{ij}\right)
$ such that

Each $\gamma_{ij} \neq 0$ for $1 \leq i \leq j \leq m$;

The following inequality holds $$ 1 -\varepsilon \geq \min_{\left[ \alpha m\right] +1\leq i\leq m}\sum_{j=\left[ \alpha m\right] +1}^{m}\gamma_{ij}^{2}\geq\varepsilon, $$ where $\left[ x\right] $ denotes the integer part of $x\in\mathbb{R}$.

- Put the last $m-[\alpha m]$ columns (by keeping the order of the column indices) of $\mathbf{T}_{m}$ into a matrix $\mathbf{R}$. Then, for the matrix $\mathbf{R}$ and a constant $\beta \in (0,\alpha)$, pick a submatrix $\mathbf{G}$ of size $[\beta m] \times (m-[\alpha m])$ from the first $[\alpha m]$ rows of $\mathbf{R}$. Also pick a submatrix of $\mathbf{H}$ of size $[\beta m] \times (m-[\alpha m])$ from the last $m-[\alpha m]$ rows of $\mathbf{R}$. Then, no such $\beta$ exists such that $\mathbf{G} = - \mathbf{H}$ or $\mathbf{G} = \mathbf{H}$ for any $\mathbf{G}$ and $\mathbf{H}$ thus picked.

The inequality in display requires that the lower right block of size $[\left( 1-\alpha\right) m]$ of $\mathbf{T}_{m}$ has rows of Euclidean norms bounded from below by $\varepsilon$ and from above by $1 -\varepsilon$ regardless of what $m$ is. It becomes highly nontrivial to obtain such a $\mathbf{T}_{m}$.

**Constructive proof for such a sequence $\left\{ \mathbf{T}_{m}\right\} _{m\geq8,m\in2\mathbb{N}}$ is preferred.**

**Note** I edited the problem. Now a third restriction is added. Without the third restriction, Walsh matrices pointed by Dr. Bill Johnson satisfy the rest two.

**Background** I encountered this problem when I was trying to contruct a counterexample to a general claim made in a published paper on the convergence of a stochastic process related to high-dimensional multiple testing.

**My attempts** The simple reflection trick to construct an orthogonal matrix is not able to obtain $\mathbf{T}_m$ for which the displayed inequality holds. I also tried (a) using the Haar measure to show the existence of such an $\mathbf{T}_m$ with positive probability but it essentially reduces to the same problem described above; (b) tried orthogonal matrix induced by dicrete cosine transform, but it can not achieve the uniformity needed in the displayed inequality.

minimumbetween the required bounds, just take a lower right block that is diagonal, with all $1$'s, except for one $\sqrt{\epsilon}$, and now it's trivial to build a unitary matrix out of this (it can be diagonal except for two non-diagonal entries). $\endgroup$ – Christian Remling Dec 6 '15 at 21:19