Equi-isoclinism from symmetry

John Jasper

Air Force Institute of Technology

(w/ Matthew Fickus, Joseph W. Iverson, Dustin G. Mixon)

The views expressed in this talk are those of the speaker and do not reflect the official policy
or position of the United States Air Force, Department of Defense, or the U.S. Government.

August 30, 2022

How do we arrange \(N\) different \(r\)-dimensional subspaces of \(\mathbb{F}^{\ }\) so that the smallest principal angle between any two is as large as possible?

\(d\)

A subspace packing problem

Q:

Equiangular Tight Frames (ETF)

Definition. Let \[\Phi = \big[\varphi_{1}\ \ \varphi_{2}\ \ \cdots\ \ \varphi_{N}\big]\in \mathbb{F}^{d\times N},\]

where each column \(\varphi_{n}\) is unit norm

\[\|\varphi_{n} \|^{2}=1.\]

1) (Tightness) \(\exists\,A>0\) such that \((\Phi^{\ast}\Phi)^{2} = A\Phi^{\ast}\Phi\).

2) (Equiangular) \(\exists\,B>0\) such that \(|\frac{1}{B}\varphi_{m}^{\ast}\varphi_{n}^{}|=1\) for \(m\neq n\).

If both 1) and 2) hold, then \(\{\varphi_{n}\}_{n=1}^{N}\) is an ETF(\(d,N)\).

\[\Phi^{\ast}\Phi = \left[\begin{array}{cccc} 1 & \varphi_{1}^{\ast}\varphi_{2} & \cdots & \varphi_{1}^{\ast}\varphi_{N}\\[1ex] \varphi_{2}^{\ast}\varphi_{1} & 1 & \cdots & \varphi_{2}^{\ast}\varphi_{N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] \varphi_{N}^{\ast}\varphi_{1} & \varphi_{N}^{\ast}\varphi_{2} & \cdots & 1\end{array}\right]\]

Equi-isoclinic tight fusion frame (EITFF)

Definition. Let \[\Phi = \big[\Phi_{1}\ \ \Phi_{2}\ \ \cdots\ \ \Phi_{N}\big]\in(\mathbb{F}^{d\times r})^{1\times N},\]

where the columns of each \(\Phi_{n}\) form ONB for a subspace (w/ dim\(=r\))

\[\Phi_{n}^{\ast}\Phi_{n} = \boldsymbol{I}.\]

1) (Tightness) \(\exists\,A>0\) such that \((\Phi^{\ast}\Phi)^{2} = A\Phi^{\ast}\Phi\).

2) (Equi-isoclinic) \(\exists\,B>0\) such that \(\frac{1}{B}\Phi_{m}^{\ast}\Phi_{n}\) is unitary for \(m\neq n\).

If both 1) and 2) hold, then \(\{\Phi_{n}\}_{n=1}^{N}\) is an EITFF(\(d,N,r)\).

\[\Phi^{\ast}\Phi = \left[\begin{array}{cccc} \boldsymbol{I} & \Phi_{1}^{\ast}\Phi_{2} & \cdots & \Phi_{1}^{\ast}\Phi_{N}\\[1ex] \Phi_{2}^{\ast}\Phi_{1} & \boldsymbol{I} & \cdots & \Phi_{2}^{\ast}\Phi_{N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] \Phi_{N}^{\ast}\Phi_{1} & \Phi_{N}^{\ast}\Phi_{2} & \cdots & \boldsymbol{I}\end{array}\right]\]

Examples

Take my favorite ETF, an ETF\((3,4)\):

\[\Phi = \frac{1}{\sqrt{3}}\left[\begin{array}{cccc} + & - & + & -\\ + & + & - & -\\ + & - & - & +\end{array}\right]\]

1)  (ETFs) \(\Phi\) is an EITFF\((3,4,1)\)

2)  (Tensors) \(\Phi\otimes\boldsymbol{I} = \dfrac{1}{\sqrt{3}}\left[\begin{array}{cc|cc|cc|cc} + &  0 & - &  0 &  + &  0 & - &  0\\ 0 &  + &  0 & - &  0 &  + &  0 & -\\ + &  0 &  + &  0 & - &  0 & - &  0\\ 0 &  + &  0 &  + &  0 & - &  0 & -\\ + &  0 & - &  0 & - &  0 &  + &  0\\ 0 &  + &  0 & - &  0 & - &  0 &  + \end{array}\right]\)

\[\exists\,\operatorname{EITFF}(d,N,r)\quad\Rightarrow\quad \exists\,\operatorname{EITFF}(md,N,mr)\]

\[\exists\,\operatorname{ETF}(d,N)\quad\Leftrightarrow\quad\exists\,\operatorname{EITFF}(d,N,1)\]

is an EITFF\((6,4,2)\)

Examples

Outline

Part I. Graph covers

Part II. Harmonic EITFFs

Part III. Totally symmetric EITFFs

Part I

Graph Covers

Signature matrix

\[\boldsymbol{S} = B\big(\Phi^{\ast}\Phi-\boldsymbol{I}\big) = \left[\begin{array}{cccc} \boldsymbol{0} & B\Phi_{1}^{\ast}\Phi_{2} & \cdots & B\Phi_{1}^{\ast}\Phi_{N}\\[1ex] B\Phi_{2}^{\ast}\Phi_{1} & \boldsymbol{0} & \cdots & B\Phi_{2}^{\ast}\Phi_{N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] B\Phi_{N}^{\ast}\Phi_{1} & B\Phi_{N}^{\ast}\Phi_{2} & \cdots & \boldsymbol{0}\end{array}\right]\]

\[= \left[\begin{array}{cccc} \boldsymbol{0} & U_{1,2} & \cdots & U_{1,N}\\[1ex] U_{2,1} & \boldsymbol{0} & \cdots & U_{2,N}\\[1ex] \vdots & \vdots & \ddots & \vdots\\[1ex] U_{N,1} & U_{N,2} & \cdots & \boldsymbol{0}\end{array}\right]\]

\[\boldsymbol{S}^{2}= B(A-2)\boldsymbol{S}+B^{2}(A-1)\boldsymbol{I}\]

\[\boldsymbol{S}^{2} = B^{2}\big((\Phi^{\ast}\Phi)^{2} - 2\Phi^{\ast}\Phi + \boldsymbol{I}\big) = B^{2}\big((A-2)\Phi^{\ast}\Phi +\boldsymbol{I}\big)\]

\[B(A-2)\big(B\Phi^{\ast}\Phi-B\boldsymbol{I}+B\boldsymbol{I}\big) + B^2\boldsymbol{I} = B(A-2)\boldsymbol{S}+B^{2}(A-1)\boldsymbol{I}\]

Suppose \(\Phi = [\Phi_{i}]_{i=1}^{N}\) is an EITFF

is called the signature matrix of the EITFF.

It's a symmetric block matrix with

• Zeros on the diagonal
• Unitaries off the diagonal
• Satisfies a quadratic:

\(\textbf{A}=\)

Graph cover example

\(\textbf{A}=\)

Graph cover example

• Zeros on diagonal
• Off-diagonal blocks are unitaries
• Satisfies a "quadratic"

\(\textbf{A}^{2}=\)

\(= 4\boldsymbol{I}_{15} + \boldsymbol{J}_{15} - (\boldsymbol{I}_{5}\otimes\boldsymbol{J}_{3})\)

\[\mathbf{A}^{2} = (N-Rc-2)\mathbf{A} + (N-1)\boldsymbol{I}_{NR} + c(\boldsymbol{J}_{N}-\boldsymbol{I}_{N})\otimes \boldsymbol{J}_{R}.\]

An \((N,R,c)\)-DRACKn is a block matrix

\[\mathbf{A} = \left[\begin{array}{cccc} \mathbf{A}_{1,1} & \mathbf{A}_{1,2} & \cdots & \mathbf{A}_{1,N}\\ \mathbf{A}_{2,1} & \mathbf{A}_{2,2} & \cdots & \mathbf{A}_{2,N}\\ \vdots & \vdots & \ddots & \vdots\\ \mathbf{A}_{N,1} & \mathbf{A}_{N,2} & \cdots & \mathbf{A}_{N,N}\end{array}\right]\]

1. \(\mathbf{A}_{i,i} = 0\)
2. \(\mathbf{A}_{i,j}\) is a permutation matrix for \(i\neq j\)
3. \(\mathbf{A}\) is symmetric
4. \(\mathbf{A}\) satisfies

Let \(\Gamma\) be the permutation group generated by \(\{\mathbf{A}_{i,j}\}_{i\neq j}\).

DRACKn's

\(\textbf{A}=\)

\[\mathbf{A}_{1,2}=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right]\cong (23)\in S_{3}\]

\[\mathbf{A}_{1,4}=\left[\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\cong (12)\in S_{3}\]

\[\mathbf{A}_{2,3}=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right]\cong (13)\in S_{3}\]

\(\Gamma = \langle \mathbf{A}_{1,4},\mathbf{A}_{1,2},\mathbf{A}_{2,3}\rangle\cong S_{3}\)             since             \(\langle (12),(23),(13)\rangle = S_{3}\)

Example with \(\Gamma\)

\(\Pi(\mathbf{A}_{1,2}) = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right]\)

\(\Pi(\mathbf{A}_{2,3}) = \left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right]\)

\(\Pi(\mathbf{A}_{1,4}) = \left[\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

The inclusion \(\Pi:\Gamma\hookrightarrow U(3)\) is a representation of \(\Gamma\)

Given a repn \(\pi\) of \(\Gamma\cong S_{3}\) we set

\(\hat{\pi}(\mathbf{{A}})=\)

Note that \(\hat{\Pi}(\mathbf{A}) = \mathbf{A}\)

\(\mathbf{S}: = \hat{\pi}(\mathbf{\hat{A}}) = \)

does satisfy a quadratic! \[\mathbf{S}^{2} = 4\boldsymbol{I}_{10}\]

Thus \(\mathbf{S}\) is the signature matrix of an EITFF\((5,5,2)\).

Theorem. (cf. Godsil,Hensel 1992) Let \(\mathbf{A}\) be an \((N,R,c)\)-DRACKn with permutation group \(\Gamma\). If \(\pi\) is any degree \(r\) constituent of the inclusion \(\Gamma\hookrightarrow U(R)\) that does not have the trivial repn as a constituent, then \(\hat{\pi}(\mathbf{A})\) is the signature matrix of an EITFF\((d,N,r)\).

Some theorems

Part II

Harmonic EITFFs

Goal: Find

Example. Let \(\mathcal{G} = \mathbb{Z}_{7} = \{0,1,2,3,4,5,6\}\),

\[\pi(g) = \begin{bmatrix} 1 & 0 & 0\\ 0 & e^{2\pi i g/7} & 0\\ 0 & 0 & e^{2 \pi i 3g/7}\end{bmatrix},\]

and \(\Phi_{0}= [1\ \ 1\ \ 1]^{\top}/\sqrt{3}\).

\(\Phi=\)

• a group \(\mathcal{G}\)
• a unitary representation \(\pi:\mathcal{G}\to U(d)\)
• an isometry \(\Phi_{0}\in\mathbb{F}^{d\times r}\)

such that \(\{\pi(g)\Phi_{0}\}_{g\in\mathcal{G}}\) is an EITFF.

Example 2. Let \(\mathcal{G} = \mathbb{Z}_{5} = \{0,1,2,3,4\}\) and \(\omega=\exp(2\pi i/5)\).

\[\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4}\end{array}\big] = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & 1 & 1\\ \cdot & 1 & i & -i & -1\end{array}\right]\]

Harmonic Tight Fusion Frames

• Abelian group: \(\mathcal{G} = \{g_{0},g_{1},\ldots,g_{N-1}\}\)
• Tight fusion frame: \(\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{\chi_{0}} & \Psi_{\chi_{1}} & \Psi_{\chi_{2}} & \cdots & \Psi_{\chi_{N-1}}\end{array}\big] \)

For each \(\chi\in\hat{\mathcal{G}}\) the matrix \(\Psi_{\chi}\) is an \(r\times D_{\chi}\) matrix. (\(D_{\chi}\) could be zero!)

Start with:

Characters: \(\hat{\mathcal{G}} = \{\chi_{0},\chi_{1},\ldots,\chi_{N-1}\}\)

\[\pi(g) := \bigoplus_{\chi\in\hat{\mathcal{G}}}\chi(g)\boldsymbol{I}_{D_{\chi}} =  \left[\begin{array}{ccccc}\hspace{-2px}\chi_{0}(g) \boldsymbol{I}_{D_{\chi_{0}}}&&&&\\&\hspace{-10px}\chi_{1}(g) \boldsymbol{I}_{D_{\chi_{1}}}&&&\\&&\hspace{-10px}\chi_{2}(g) \boldsymbol{I}_{D_{\chi_{2}}}&&\\&&&\hspace{-15px}\ &\\&&&&\hspace{-10px}\chi_{N-1}(g) \boldsymbol{I}_{D_{\chi_{N-1}}}\hspace{-2px}\end{array}\right]\]

Harmonic TFF generated by \(\Psi\): 

\[\Phi \doteq\left[\begin{array}{c|c|c|c|c} \pi(g_{0})\Psi^{\ast} & \pi(g_{1})\Psi^{\ast} & \pi(g_{2})\Psi^{\ast} & \cdots & \pi(g_{N-1})\Psi^{\ast}\end{array}\right]\]

\(\ddots\)

Cross-Grams

Harmonic TFF generated by \(\Psi\): 

\[\Phi \doteq \left[\begin{array}{c|c|c|c|c} \pi(g_{0})\Psi^{\ast} & \pi(g_{1})\Psi^{\ast} & \pi(g_{2})\Psi^{\ast} & \cdots & \pi(g_{N-1})\Psi^{\ast}\end{array}\right]\]

Set \(\Phi_{g} \doteq \pi(g)\Psi^{\ast}\).

\[\Phi_{g}^{\ast}\Phi_{h}^{} \doteq (\pi(g)\Psi^{\ast})^{\ast}(\pi(h)\Psi^{\ast}) = \Psi\pi(g^{-1})\pi(h)\Psi^{\ast} \doteq \Phi_{g_{0}}^{\ast}\Phi_{h^{-1}g}^{}\]

Cross-Grams:

The Gram matrix is block circulant.

Also,

\[\Phi_{g_{0}}^{\ast}\Phi_{g} \doteq \sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\Psi_{\chi}\Psi_{\chi}^{\ast} = \sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\mathbf{P}_{\chi}=:\mathbf{M}_{g}\]

\(\mathbf{P}_{\chi}\) is the projection onto \(\operatorname{ran}\Psi_{\chi}\)

\(\mathbf{M}_{g}\) is the entrywise Fourier transform of the \(\mathbf{P}_{\chi}\)'s.

Harmonic EITFFs

Theorem (Fickus,Iverson,J,Mixon '21) Suppose

• \(\mathcal{G}\) is a finite abelian group
• \(\Psi=[\Psi_{\chi}]_{\chi\in\mathcal{\hat{G}}}\) is a TFF
• \(\Phi = [\Phi_{g}]_{g\in\mathcal{G}}\) is the harmonic frame generated by \(\Psi\)

Then,

• \(\Phi\) is a TFF
• \(\Phi\) is an EITFF if and only if  \[\mathbf{M}_{g}:=\sum_{\chi\in\hat{\mathcal{G}}}\chi(g)\Psi_{\chi}\Psi_{\chi}^{\ast} \]  are a common multiple of unitaries for \(g\in \mathcal{G}\setminus\{g_{0}\}\). 

Example 2 (redux). Let \(\mathcal{G} = \mathbb{Z}_{5} = \{0,1,2,3,4\}\) and \(\omega=\exp(2\pi i/5)\).

\[\Psi = \big[\begin{array}{c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4}\end{array}\big] = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & 1 & 1\\ \cdot & 1 & i & -i & -1\end{array}\right]\]

\[=\frac{1}{2}\left[\begin{array}{cc}-1 & \omega^{-k}-i\omega^{-2k}+i\omega^{-3k}-\omega^{-4k}\\\omega^{-k}+i\omega^{-2k}-i\omega^{-3k}-\omega^{-4k} & -1\end{array}\right]\]

\[\mathbf{P}_{0} = \begin{bmatrix}0&0\\0&0\end{bmatrix}\]

\[\mathbf{P}_{1} = \frac{1}{2}\begin{bmatrix}1&1\\1&1\end{bmatrix}\]

\[\mathbf{P}_{2} = \frac{1}{2}\begin{bmatrix}1&-i\\i&1\end{bmatrix}\]

\[\mathbf{P}_{3} = \frac{1}{2}\begin{bmatrix}1&i\\-i&1\end{bmatrix}\]

\[\mathbf{P}_{4} = \frac{1}{2}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\]

\[\mathbf{M}_{k} = \sum_{j=0}^{4}\omega^{-jk}\mathbf{P}_{j}\]

Indeed, \(\mathbf{M}_{k}^{\ast}\mathbf{M}_{k}^{} = \frac{3}{2}\boldsymbol{I}_{2}\) for each nonzero \(k\in\mathbb{Z}_{5}\).

Therefore \(\Phi\) is a EITFF\((4,5,2)\) (five two-dimensional subspaces of \(\mathbb{C}^{4}\))

Example 1 (redux). Let \(\mathcal{G} = \mathbb{Z}_{7} = \{0,1,2,3,4,5,6\}\) and \(\omega=\exp(2\pi i/7)\)

\[\Psi = \big[\begin{array}{c|c|c|c|c|c|c} \Psi_{0} & \Psi_{1} & \Psi_{2} & \Psi_{3} & \Psi_{4} & \Psi_{5} & \Psi_{6}\end{array}\big] = \big[\begin{array}{c|c|c|c|c|c|c} 1 & 1 & \cdot & 1 & \cdot & \cdot & \cdot\end{array}\big]\]

\[\mathbf{P}_{0} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{1} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{2} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{3} = \begin{bmatrix}1\end{bmatrix},\ \mathbf{P}_{4} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{5} = \begin{bmatrix}0\end{bmatrix},\ \mathbf{P}_{6} = \begin{bmatrix}0\end{bmatrix}\]

\[\mathbf{M}_{k} = \omega^{0k}+\omega^{-k}+\omega^{-3k}\]

\(\mathbf{M}_{k}\) has the same modulus for all \(k\neq 0\)

The Fourier transform of \([1\ \ 1\ \ 0\ \ 1\ \ 0\ \ 0\ \ 0]^{\top}\) is "spike+flat"

\(\{0,1,3\}\) is a difference set in \(\mathbb{Z}_{7}\).

\(\Updownarrow\)

\(\Updownarrow\)

New Harmonic EITFFs

Theorem. (Fickus, Iverson, J, Mixon '21) Write \(\mathbb{F}_{q}^{\times} = \langle \alpha\rangle = \{\alpha^{j}\}_{j=0}^{q-2}\)

1. If \(q\) is an odd prime power, and \(\chi\) is an additively odd multiplicitive character, then the harmonic TFF generated by \[\Psi = \frac{1}{\sqrt{2}}\left[\begin{array}{c|c|c|c|c}\cdot & 1 & 1 & \cdots & 1\\ \cdot & \chi(\alpha^{0}) & \chi(\alpha^{1}) & \cdots & \chi(\alpha^{q-2})\end{array}\right]\] is an EITFF\((q-1,q,2)\). (\(19\leq q\equiv 1\mod 4\) seem to be new.)
2.  If \(q\geq 4\) is any prime power and \(\chi\neq 1\) is a multiplicative character of \(\mathbb{F}_{q}\), then the harmonic TFF generated by \[\Psi = \frac{1}{\sqrt{2q-2}}\left[\begin{array}{c|c|c|c} \sqrt{2q-2} & \sqrt{q-2} & \cdots & \sqrt{q-2}\\  0 & \sqrt{q}\chi(\alpha^{0}) & \cdots & \sqrt{q}\chi(\alpha^{q-2})\end{array}\right]\] is an EITFF\((q,q,2)\). (\(q\equiv 3\mod 4\) seem to be new.)

Part III

Totally Symmetric EITFFs

Subspaces \(\mathcal{W}=\{W_{i}\}_{i=1}^{N}\) in \(\mathbb{R}^{d}\) have a group of symmetries

\[\operatorname{Sym}(\mathcal{W}): = \{\text{permutations of }\mathcal{W}\text{ by orthogonal maps}\}.\]

Goal: Find totally symmetric subspaces, i.e., \(\operatorname{Sym}(\mathcal{W}) = S_{N}\).

Plan:

Totally symmetric subspaces

"Symmetry encourages optimality"

• Take a representation \(\pi:S_{N}\to O(d)\)
• Let \(W\) be an invariant subspace of \(\pi|_{\operatorname{stab}(N)}\)
• Let \(\mathcal{W} = \pi(S_{N})\cdot W\)

Representations of \(S_{N}\)

Irr Reps of \(S_{N}\qquad\qquad\qquad\) Young diagrams

\(1-1\)

\(\longleftrightarrow\)

\(\pi\cong\)

\(\Rightarrow\)

\(\pi|_{\operatorname{stab}(N)}\cong\)

\(\oplus\)

\(\oplus\)

Each summand in \(\pi|_{\operatorname{stab}(N)}\) gives a choice of \(W\).

Diagram + Extremal cell = totally symmetric subspaces

When are they equi-isoclinic?

Some totally symmetric EITFFs

\((5,5,2)\)

\((14,7,5)\)

\((16,6,6)\)

\((90,8,20)\)

\((42,9,14)\)

\((210,10,42)\)

\((168,9,56)\)

\((448,10,70)\)

Using the Hook length formula we calculate \((d,N,r)\):

MOAR boxes!

\(\pi:=\pi_{1}\oplus\pi_{2}\)   where

\(\pi_{1}\cong\)

,         \(\pi_{2}\cong\)

\(\rho\cong\)

is a constituent of both \(\pi_{1}|_{\operatorname{stab}(N)}\) and \(\pi_{2}|_{\operatorname{stab}(N)}\)

\(\Rightarrow\) \(r\)-dim \(\rho\)-invariant subspaces \(W_{1}\subset\mathbb{R}^{d_{1}}\) and \(W_{2}\subset\mathbb{R}^{d_{2}}\)

\(\rho\) acts on \(W_{1}\) and \(W_{2}\) simultaneously relative to the ONBs

\[\{b_{1,i}\}_{i=1}^{r}\subset W_{1}\quad\text{and}\quad \{b_{2,i}\}_{i=1}^{r}\subset W_{2}\]

Set \(W:=\operatorname{span}\{\sqrt{d_{1}}b_{1,i}\oplus \sqrt{d_{2}}b_{2,i}\}_{i=1}^{r}\).

\(\pi(S_{N})\cdot W\) is a totally symmetric TFF.

Can we take \(\pi\) to be reducible? Yes! For example:

MOAR boxes!

"One red box": See previous examples.

"Two red boxes":

\(\oplus\)

\((42900, 13, 7700)\)

\(\oplus\)

\((∼ 10^{11}, 25, ∼ 10^{8})\)

"Three red boxes":   \(N\sim 7\cdot 10^{12},\ 10^{10^{13}}<r<d<10^{10^{14}}\)

(found using CAD \(+\) group law)

Ideas for future research

1. Are there more DRACKn's with nonabelian permutations groups \(\Gamma\)?
2. More harmonic EITFFs? (We found a harmonic EITFF(11,11,3), but it was...complicated)
3. Are there totally symmetric EITFFs from "four red boxes"? "\(N\) red boxes"?