### N EGATIVE C LIQUES IN S ETS OF

### E QUIANGULAR L INES

Emily J. King

joint work with Matt Fickus, John Jasper, Dustin Mixon, and Xiaoxian Tang

University of Bremen

Tight Frames and Approximation February 20–23, 2018

### O UTLINE

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^{OUNDS ON}

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^{EAL}

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MBEDDED IN### ETF

S### O UTLINE

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S3/35

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ? 1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.)

2. The inner products between the vectors are 2A. optimallysmalland

2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)

3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.)

How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

### O RTHONORMAL B ASES

LetΦ={*ϕ*_{j}}be an orthonormal basis for**F**^{d}for**F**=_{R,}_{C.}

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)
3. For anyx∈**F**^{d},x=_{∑}_{j}hx,*ϕ*_{j}i*ϕ*_{j}.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling)

### E QUIANGULAR L INES D

^{EFINITION}

Let**F**=**C**or**R. Let**Φ={*ϕ*_{j}}^{n}_{j=1}⊂**F**^{k}with
ϕ_{j}

=1 for all
j∈ {1, . . . ,n}. If there exists an*α*such that for allj6=`,

h*ϕ*_{j},*ϕ*_{`}i^{}=*α,*
Φis a set ofequiangular lines.

If further for allx∈**F**^{k}

### ∑

n j=1|hx,*ϕ*_{j}i|^{2}= ^{n}

kkxk^{2} ⇔ x= ^{n}
k

### ∑

j

hx,*ϕ*_{j}i*ϕ*_{j},
thenΦis anequiangular tight frame (ETF).

By slight abuse of notation,

I Φ= *ϕ*_{1} *ϕ*2 . . . *ϕ*n
, and

I *α*is the “angle.”

### T

^{HEOREM}

(Goyal, Kovaˇcevi´c, Kelner 2001; Strohmer, Heath 2003; Benedetto, Kolesar 2006) ETFs are optimally robust against erasures and noise.

5/35

### E XAMPLES IN **R**

^{2}

I ONB, ETF

I ETF, maximal set of equiangular lines

I FUNTF, worst case coherence

I Equiangular lines which are not an ETF

### R ESEARCH Q UESTIONS

Q1 : Givend(and 0<*α*<1), what is themaximal sizes(d)(resp.,
s* _{α}*(d)) of a set of equiangular lines (resp., with angle

*α) in*

**R**

^{d}?

Q2 : Given a specific ETF or class of ETFs, what is thestructure of linear dependenciesof the vectors?

7/35

### G RAM M ATRICES

Instead ofΦ, we will usually deal with theGram matrix
G(Φ) =Φ^{∗}Φ.

LetInbe then×nidentity andJnthen×nall-ones matrix (where we writeIandJwhen clear from context).

Basic linear alg: IfG= (a−b)In+bJn, thenG
has a simple eigenvalue*λ*_{1}=a+ (n−1)band
an eigenvalue*λ*_{2}=a−bwith multiplicityn−1.

a b . . . b b a . . . b ... ... . .. ...

b b . . . a

### G RAM M ATRICES , II

I

1 0

0 1

I

1 −^{1}_{2} −^{1}_{2}

−^{1}_{2} 1 −^{1}_{2}

−^{1}_{2} −^{1}_{2} 1

I

1 0 −1 0

0 1 0 −1

−1 0 1 0

0 −1 0 1

I 1

√ 2

√ 2 2

2 1

!

9/35

### S WITCHING E QUIVALENCE

Basic linear algebra:G(_{Φ}) =G(_{Ψ}) ⇔ _{Φ}=_{UΨ}for some unitaryU.

GivenΦ={*ϕ*1, . . . ,*ϕ*n}, ˜Φ={*ϕ*n, . . . ,−*ϕ*1}has the same geometric
and linear algebraic properties.

### D

^{EFINITION}

Two sets of unit vectorsΦandΨin**F**^{d}areswitching equivalent,
denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm
diagonal entries and a permutation matrixCsuch that

(BC)·G(_{Φ})·(BC)^{−1} = G(_{Ψ})_{.}

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)^{−1}=_{Ψ.}

=((Van Lindt & Seidel 1966+generalization to**F) +**permutations) =
(Projective unitary equivalence+permutations)

### S WITCHING E QUIVALENCE

Basic linear algebra:G(_{Φ}) =G(_{Ψ}) ⇔ _{Φ}=_{UΨ}for some unitaryU.

GivenΦ={*ϕ*1, . . . ,*ϕ*n}, ˜Φ={*ϕ*n, . . . ,−*ϕ*1}has the same geometric
and linear algebraic properties.

### D

^{EFINITION}

Two sets of unit vectorsΦandΨin**F**^{d}areswitching equivalent,
denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm
diagonal entries and a permutation matrixCsuch that

(BC)·G(_{Φ})·(BC)^{−1} = G(_{Ψ})_{.}

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)^{−1}=_{Ψ.}

=((Van Lindt & Seidel 1966+generalization to**F) +**permutations) =
(Projective unitary equivalence+permutations)

10/35

### S WITCHING E QUIVALENCE

Basic linear algebra:G(_{Φ}) =G(_{Ψ}) ⇔ _{Φ}=_{UΨ}for some unitaryU.

GivenΦ={*ϕ*1, . . . ,*ϕ*n}, ˜Φ={*ϕ*n, . . . ,−*ϕ*1}has the same geometric
and linear algebraic properties.

### D

^{EFINITION}

Two sets of unit vectorsΦandΨin**F**^{d}areswitching equivalent,
denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm
diagonal entries and a permutation matrixCsuch that

(BC)·G(_{Φ})·(BC)^{−1} = G(_{Ψ})_{.}

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)^{−1}=_{Ψ.}

=((Van Lindt & Seidel 1966+generalization to**F) +**permutations) =

### T RIPLE P RODUCTS

### T

^{HEOREM}

(Godsil and Royle 2001; Chien and Waldron 2016) LetΦ,Ψ⊂**F**^{d}with

|Φ|=|Ψ|=n be equiangular. ThenΦ∼=Ψif and only if there exists a
*σ*∈Snsuch that for all i6=j6=k6=i.

h*ϕ*_{i},*ϕ*_{j}ih*ϕ*_{j},*ϕ*_{k}ih*ϕ*_{k},*ϕ*_{i}i=h*ψ** _{σ(i)}*,

*ψ*

*ih*

_{σ(j)}*ψ*

*,*

_{σ(j)}*ψ*

*ih*

_{σ(k)}*ψ*

*,*

_{σ(k)}*ψ*

*i. When*

_{σ(i)}**F**=

**R**and ignoring permutations, this gives precisely the structure of thetwo-graphwhich represents equivalence classes of switching equivalences. (With permutations, get isomorphisms of the two-graphs.)

11/35

### N EGATIVE C LIQUES

### D

^{EFINITION}

LetΦbe a set of equiangular lines with angle*α. If*X⊂Φis such that
X∼=YwithG(Y) = (1+*α*)I−*αJ, then we call*Xanegative clique.

I (K, Tang 2016) LetXbe a maximal negative clique in a givenΦ.

Xis called aK-base.

I (Fickus, Jasper, K, Mixon 2017) IfXis a negative clique of size 1+ (1/α), we callXa1/α-regular simplex.

Negative cliques have size≤1+1/α.

When the bound is saturated, they form a tight frame for their span (Fickus, Jasper, K, Mixon 2017);

otherwise they are linearly independent (e.g., Lemmens & Seidel 1973).

### N EGATIVE C LIQUES

### D

^{EFINITION}

LetΦbe a set of equiangular lines with angle*α. If*X⊂Φis such that
X∼=YwithG(Y) = (1+*α*)I−*αJ, then we call*Xanegative clique.

I (K, Tang 2016) LetXbe a maximal negative clique in a givenΦ.

Xis called aK-base.

I (Fickus, Jasper, K, Mixon 2017) IfXis a negative clique of size 1+ (1/α), we callXa1/α-regular simplex.

Negative cliques have size≤1+1/α.

When the bound is saturated, they form a tight frame for their span (Fickus, Jasper, K, Mixon 2017);

otherwise they are linearly independent (e.g., Lemmens & Seidel 1973).

12/35

### O UTLINE

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QUIANGULAR### L

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^{OUNDS ON}

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^{EAL}

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IMPLICES### E

MBEDDED IN### ETF

S### P ILLAR D ECOMPOSITION

### D

EFINITION(Lemmens & Seidel 1973) LetΦ∈_{R}^{d}be equiangular withK-baseX.

LetΞdenote the subspace spanned byX. Elements ofΦwhich lie in
the same coset ofΞ^{⊥}are called pillars.

### P

ROPOSITION(Lemmens & Seidel 1973; K, Tang 2016) Let*ϕ*∈Φ\X. If any K-base is of
size1+1/α, then the norm of P_{Ξ}⊥*ϕ*is equal to*α. If any K-base is of size*

<1+1/α, then the norm of P_{Ξ}⊥*ϕ*depends on the number of negative
inner productsh*ϕ,*x_{i}i, x_{i}∈X.

14/35

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper
bounds ons*α*(d)for*α*the reciprocal of an odd integer between 5
and a√

2d+1. (3 solved.)

I For each possible*α, we consider all of the possible sizes of*
K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper
bounds ons*α*(d)for*α*the reciprocal of an odd integer between 5
and a√

2d+1. (3 solved.)

I For each possible*α, we consider all of the possible sizes of*
K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

15/35

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper
bounds ons*α*(d)for*α*the reciprocal of an odd integer between 5
and a√

2d+1. (3 solved.)

I For each possible*α, we consider all of the possible sizes of*
K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

### B ASIC P ROCEDURE OF K, T ANG 2016

*α*(d)for*α*the reciprocal of an odd integer between 5
and a√

2d+1. (3 solved.)

I For each possible*α, we consider all of the possible sizes of*
K-bases. (≥2,≤1+ (1/α)).

15/35

### S PHERICAL T WO -D ISTANCE S ETS

### D

EFINITIONLetΦ={*ϕ*_{1}, . . . ,*ϕ*n} ⊂_{R}^{d}be a set of unit normed vectors and let
*α,β*∈_{R. The}_{Φ}is aspherical two-distance setifh*ϕ*_{i},*ϕ*_{j}i ∈ {*α,β*}for
alli,j∈ {1, . . . ,n},i6=j. We denote bys(d,*α,β*)the largest size of a
spherical two-distance set with the given parameters.

Note: An equiangular set of lines is a spherical two-distance set w.r.t
*α,*−*α, and thus*s*α*(d) =s(d,*α,*−*α*).

### N EW U PPER B OUND ON s

_{1/5}

### ( _{d} )

### T

HEOREM(K, Tang 2016) LetΦ⊂**R**^{d}be an equiangular set with angle1/5. If
d>60, then

|Φ| ≤ 148+3·s(d, 1/13,−5/13) ≤ 148+^{648d}(d+2)
47d+169 .

17/35

### A TASTE OF THE CASES T

^{HEOREM}

(K, Tang 2016) For n=1, . . . ,bK/2cand for each equivalence class
x⊂X(_{K,}_{n}), we have the following upper bounds on|x|.

(1) If n=1, then

|x| ≤

(r−K, 1≥K−^{(1/α)+1}_{2}

1−α

l(K,1)−α, 1<K−^{(1/α)+1}_{2} .
(2) If1<n<K−^{(1/α)+1}_{2} , then|x| ≤ r+1.

(3) If n=K−^{(1/α)+1}_{2} , then|x| ≤ r−K+b2α^{r−K}_{1−α}c.
(4) If K−^{(1/α)+1}_{2} <n<b^{K}_{2}c, then

|x| ≤ s(r,*β,γ*),
where*β*= ^{α}_{1−l(K,n)}^{−l(K,n)}and*γ*= ^{−}_{1−l(K,n)}^{α}^{−l(K,n)}.

### E QUIANGULAR L INES IN **R**

^{d}

### T

HEOREM### (K, T

ANG### 2016)

Let m be the largest positive integer such that(2m+1)^{2}≤d+2. Then

s(d) ≤

4d(m+1)(m+2)

(2m+3)^{2}−d , d=44, 45, 46, 76, 77, 78, 117, 118, 166,
222, 286, 358

(^{(2m+1)}^{2}^{−2})(^{(2m+1)}^{2}^{−1})

2 , other k between44and400

.

Applied the SDP approach of [Bachoc, Valentin 2008; Barg, Yu 2014]

to bound the size of certain spherical two distance sets in the cases they arose.

19/35

### N EW B OUND VS . O LD

FIGURE:K, Tang 2016 and**sdp**(d,^{1}_{5},−^{1}_{5})

— :^{d(d+1)}_{2} ? ? ?:**sdp(d,**^{1}_{5},−^{1}_{5}) +++: K, Tang
2016

FIGURE:K, Tang 2016 and**sdp**(d,^{1}_{7},−^{1}_{7})

— :^{k(k+1)}_{2} ? ? ?:**sdp(d,**^{1}_{7},−^{1}_{7}) +++: K, Tang
2016

### P OSITIVE C LIQUES

One may similarly define apositive clique.

For all 0≤*α*<1 there exist at leastdvectors In
**F**^{d}with pairwise inner product*α*

Geometrically, one may think of “pushing”

vectors in an onb together.

One may analyze projections onto orthogonal complements of positive cliques and use Ramsey theory to obtain asymptotic relative bounds. (Balla, Dr¨axler, Keevash, Sudakov 2018)

21/35

### O UTLINE

### E

QUIANGULAR### L

INES### B

^{OUNDS ON}

### R

^{EAL}

### E

^{QUIANGULAR}

### L

^{INES}

### S

IMPLICES### E

MBEDDED IN### ETF

S### B INDERS

### D

^{EFINITION}

### (F

^{ICKUS}

### , J

^{ASPER}

### , K, M

^{IXON}

### 2017)

LetΦbe an ETF. The set of subsets of vectors (or the corresponding incidence matrix) which are 1/α-regular simplices is thebinder.

These are the smallest sets of linearly dependent vectors in the ETF:

For a general set of unit vectorsΦ,

size of the smallest set of linearly dependent vectors inΦ ≥ 1+ ^{1}
*µ*(_{Φ})^{.}
(Gerschgorin circle theorem applied to the Gram matrix. Donoho,
Elad 2003)

23/35

### B INDER F INDER

BinderFinderis a relatively short Matlab code that uses triple

products and some clever combinatorial tricks to compute the binder of a given ETF.

(Could also be used on sets of equiangular lines.)

Code available for download at:

http://www.math.uni-bremen.de/cda/

### B INDERS OF ETF S IN **C**

^{3}

^{×}

^{9}

Perhaps the first investigation of linear dependencies in equiangular
Gabor frames (SIC-POVMs) was presented in a talk by Hughston in
2007 (cited in Dang, Blanchfield, Bengtsson, Appleby 2013), where
the linear dependencies of certain SIC-POVMs in**C**^{3}were shown to
be represented by the Hesse configuration.

I came to the question via the construction of ETFs in (Jasper, Mixon, Fickus 2013). This construction involves a tensor-like construction of an incidence matrix of a BIBD with a 1/α-regular simplex. (Bad algebraic spread, butgoodgeometric spread?!?!?)

25/35

### B INDERS OF ETF S IN **C**

^{3}

^{×}

^{9}

Perhaps the first investigation of linear dependencies in equiangular
Gabor frames (SIC-POVMs) was presented in a talk by Hughston in
2007 (cited in Dang, Blanchfield, Bengtsson, Appleby 2013), where
the linear dependencies of certain SIC-POVMs in**C**^{3}were shown to
be represented by the Hesse configuration.

I came to the question via the construction of ETFs in (Jasper, Mixon, Fickus 2013). This construction involves a tensor-like construction of an incidence matrix of a BIBD with a 1/α-regular simplex. (Bad algebraic spread, butgoodgeometric spread?!?!?)

### H ESSE C ONFIGURATION

The Hesse configuration is the set of all lines in**F**^{2}_{3}:

FIGURE: By David Eppstein - Own work, CC0,

https://commons.wikimedia.org/w/index.php?curid=18920067

26/35

### “N ORMAL ” C ONFIGURATION

Let*ζ*=e^{2πi/3},*θ*∈[0, 2π/6]\{0, 2π/9}.

SIC-POVMs:(Hughston 2007; Dang, Blanchfield, Bengtsson, Appleby 2013)

0 0 0 −e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2} 1 1 1

1 1 1 0 0 0 −e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2}

−e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2} 1 1 1 0 0 0

Kirkman ETFs(Fickus, Jasper, Mixon 2013):

BIBD(3, 2, 1) D_{3}

0 1 1 1 0 1 1 1 0

&

1 1 1

1 *ζ* *ζ*^{2}
1 *ζ*^{2} *ζ*

=:

w0

w1

w_{2}

⇒

0 0 0 w0 w1

w1 0 0 0 w0

w w 0 0 0

### “N ORMAL ” C ONFIG . B INDER & G RAM OF B INDER

Left: Binder of a “normal” SIC-POVM in**C**^{3}, Right: The Gram matrix
of the binder.

28/35

### O BTAINING THE H ESSE C OFIGURATION

Let*ζ*=e^{2πi/3},*θ*∈ {0, 2π/9}.

SIC-POVMs:(Hughston 2007; Dang, Blanchfield, Bengtsson, Appleby 2013)

0 0 0 −e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2} 1 1 1

1 1 1 0 0 0 −e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2}

−e^{iθ} −e^{iθ}*ζ* −e^{iθ}*ζ*^{2} 1 1 1 0 0 0

Polyphase BIBD ETFs(Fickus, Jasper, Mixon, Peterson, Watson 2017;

Fickus, Jasper, K, Mixon 2017):

BIBD(3, 2, 1) D_{3}

0 1 1 1 0 1 1 1 0

&

1 1 1

1 *ζ* *ζ*^{2}
1 *ζ*^{2} *ζ*

=:

w_{0}
w_{1}
w2

⇒

0 0 0 w_{0} −w_{1}

−w_{1} 0 0 0 w_{0}

### H ESSE C ONFIGURATION AS A B INDER

Left: Hesse Configuration binder of a SIC-POVM in**C**^{3}, Right: The
Gram matrix of the binder.

30/35

### B INDER OF H OGGAR ’ S L INES

Left: Binder of Hoggar’s lines (non-Gabor SIC POVM in**C**^{8}with 1008
simplices), Right: The Gram matrix of the binder.

### C ONCLUSION

I Fond memories (nightmares?) of using Sylowp-groups in the classification of finite simple groups.

I

32/35

### M ISSING C OMRADES

## Thanks to Shayne for

## organizing/chauffeuring and to New Zealand for being so beautiful!

http://www.math.uni-bremen.de/cda/

“New Upper Bounds for Equiangular Lines by Pillar Decomposition”

on arXiv (the paper formerly known as “Computing Upper Bounds for Equiangular Lines in Euclidean Space”)

“Equiangular tight frames that contain regular simplices” on arXiv

34/35