New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$

Abstract: In this paper we provide new constructions of equiangular tight frames of size $2d$ in $\mathbb{C}<sup>d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3\pmod 4$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1\pmod 4$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.

• The paper introduces ETF-amicable pairs, a Kronecker-product framework that combines complex Hadamard matrices with ETF signature matrices to generate new equiangular tight frames.
• The constructions produce previously unknown optimal packings, including explicit ETFs in dimensions 93 and 105, and provide infinite families linked to Paley conference matrices and amicable Hadamard pairs.
• The paper extends Turyn’s power construction to ETF cores, while proving that the multiplying method is restricted to signature parameters c=0 or c=±2 and does not resolve the general doubling conjecture.

This paper develops new multiplicative constructions of equiangular tight frames (ETFs) of size $d\times 2d$ in $\mathbb{C}^d$, generalizing both the doubling construction of Fallon and Iverson and Turyn's power construction for conference matrices. The central object is a new notion of an "ETF-amicable pair" of matrices, which couples a Hermitian complex Hadamard matrix with a signature matrix of an ETF; the paper shows such pairs arise from amicable Hadamard pairs, from explicit one-parameter families of order 6, and from Paley conference matrices, yielding many previously unknown dimensions in which optimal line packings exist.

Background: ETFs and signature matrices

An ETF of size $d\times N$ is a set of $N$ unit vectors in $\mathbb{C}^d$ attaining the Welch bound on coherence,

$|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$

and thus constitutes an optimal packing of $N$ lines in $\mathbb{CP}^{d-1}$. Following Holmes and Paulsen, an ETF is encoded by its signature matrix $S$, defined by $G = I + \alpha S$ where $\mathbb{C}^d$0 is the Gram matrix and $\mathbb{C}^d$1 the Welch-bond coherence. The Holmes–Paulsen criterion states that $\mathbb{C}^d$2 is a signature matrix of a $\mathbb{C}^d$3 ETF if and only if $\mathbb{C}^d$4 has zero diagonal and unimodular off-diagonal entries and satisfies

$\mathbb{C}^d$5

The case $\mathbb{C}^d$6 forces $\mathbb{C}^d$7, so $\mathbb{C}^d$8, connecting these ETFs directly to Hadamard and conference matrices. The Fallon–Iverson conjecture asserts that a $\mathbb{C}^d$9 ETF exists for every $d\times N$0; it has been verified computationally up to $d\times N$1, with explicit constructions known for all $d\times N$2 except $d\times N$3.

The Kronecker multiplication construction

The main structural contribution is the notion of an ETF-amicable pair: a pair $d\times N$4 of $d\times N$5 matrices where $d\times N$6 is a Hermitian complex Hadamard matrix, $d\times N$7 is a signature $d\times N$8-matrix, and

$d\times N$9

for some real constant $N$0. The key theorem states that if $N$1 is ETF-amicable with parameter $N$2 and $N$3 is any signature $N$4-matrix, then

$N$5

is a signature $N$6-matrix. The proof is a direct verification of the quadratic condition using $N$7, $N$8, and $N$9.

For order-2 amicable pairs, this construction recovers exactly the doubling construction of Fallon and Iverson, which produces an $\mathbb{C}^d$0 ETF from any ETF with $\mathbb{C}^d$1. The author notes that the apparent extra freedom in choosing the order-2 pair collapses under Seidel equivalence: different pairs with the same value of $\mathbb{C}^d$2 yield switching-equivalent frames.

Consequences via amicable Hadamard pairs

Taking $\mathbb{C}^d$3, the construction gives: if $\mathbb{C}^d$4 is a Hermitian complex Hadamard matrix of order $\mathbb{C}^d$5 and $\mathbb{C}^d$6 is the signature matrix of an $\mathbb{C}^d$7 ETF with $\mathbb{C}^d$8, then existence of a $\mathbb{C}^d$9 ETF implies existence of a $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$0 ETF. The anticommutation condition $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$1 is supplied by amicable pairs of Hadamard matrices — a symmetric Hadamard matrix paired with a skew Hadamard matrix satisfying $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$2. Setting $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$3 and $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$4 (where $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$5 is skew) verifies the condition algebraically. Consequently:

• Amicable Hadamard pairs of order $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$6 exist for all prime powers $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$7, so whenever a $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$8 ETF exists, so does one at dimension multiplied by $|\langle x_i,x_j\rangle|^2 \ge \frac{N-d}{d(N-1)},$9.
• More generally, by Seberry's catalogue of amicable pairs, multiplication factors include $N$0, $N$1 for prime powers $N$2, $N$3 when circulant Hadamard cores of order $N$4 exist, and arbitrary products thereof.

Since complex Hadamard matrices exist in every order but amicable real pairs do not, the restriction to amicable pairs is a genuine constraint inherited from the classical design-theoretic setting; conjecturally, amicable pairs exist for all orders divisible by 4, which would make the multiplier construction essentially unconditional.

Explicit families

Two explicit families of ETF-amicable pairs are presented, both found initially with computational/AI assistance and then verified or generalized analytically.

Order 6. A one-parameter family $N$5, $N$6, built on Et-Taoui's family of complex conference matrices of order 6 satisfies $N$7, $N$8, and $N$9. Combined with Strohmer's ETFs of signature $\mathbb{CP}^{d-1}$0 arising from skew Hadamard matrices of order $\mathbb{CP}^{d-1}$1, this yields a $\mathbb{CP}^{d-1}$2 ETF whenever a skew Hadamard matrix of order $\mathbb{CP}^{d-1}$3 exists. Via Paley's construction this confirms the doubling conjecture for all $\mathbb{CP}^{d-1}$4 with $\mathbb{CP}^{d-1}$5; notably, it supplies an explicit construction at $\mathbb{CP}^{d-1}$6, one of the four gaps below dimension 150 in prior tables. The remaining gap $\mathbb{CP}^{d-1}$7 follows from Goethals and Seidel's skew Hadamard matrix of order 36.

Orders $\mathbb{CP}^{d-1}$8 for $\mathbb{CP}^{d-1}$9. For the symmetric Paley conference matrix $S$0 of order $S$1, indexed by the projective line $S$2, the author constructs a fixed-point-free involution $S$3 (for nonsquare $S$4) together with signs $S$5, producing a signed permutation matrix $S$6 with $S$7, $S$8, and $S$9. Setting $G = I + \alpha S$0 yields a Hermitian complex Hadamard matrix anticommuting with $G = I + \alpha S$1. This gives a $G = I + \alpha S$2 ETF from any $G = I + \alpha S$3 ETF. The author remarks that the underlying mechanism — a Hermitian monomial involution $G = I + \alpha S$4 with $G = I + \alpha S$5 applied as $G = I + \alpha S$6 — plausibly extends to other structured frames such as harmonic frames, though this is left unverified.

The power construction

The second half of the paper adapts Turyn's power construction for conference matrices to ETF cores. A core $G = I + \alpha S$7 of a $G = I + \alpha S$8 ETF (the standard-form signature matrix with first row and column removed) is characterized by: Hermitian, zero diagonal, unimodular off-diagonal entries, $G = I + \alpha S$9, and $\mathbb{C}^d$00.

For $\mathbb{C}^d$01, the core $\mathbb{C}^d$02 of a $\mathbb{C}^d$03 ETF is built from a core of a $\mathbb{C}^d$04 ETF. For general exponents, define strings over the alphabet $\mathbb{C}^d$05 in which every $\mathbb{C}^d$06 is followed by a $\mathbb{C}^d$07 and vice versa (cyclically), and set $\mathbb{C}^d$08 over all such strings of length $\mathbb{C}^d$09, where $\mathbb{C}^d$10 is the corresponding Kronecker product. The combinatorial heart of the proof is a uniqueness lemma: every 0/1-sequence of length $\mathbb{C}^d$11 (except the all-ones sequence) corresponds to exactly one string, which forces cross terms $\mathbb{C}^d$12 to vanish for distinct strings and makes cancellation of unwanted $\mathbb{C}^d$13-products exact. The result is that $\mathbb{C}^d$14, giving:

If a $\mathbb{C}^d$15 ETF exists, then a $\mathbb{C}^d$16 ETF exists for every $\mathbb{C}^d$17.

Combined with Paley conference matrices, this covers dimensions of the form $\mathbb{C}^d$18 beyond what doubling alone provides.

Limitations and open questions

The paper is candid about the scope of its methods. A spectral argument shows that ETF-amicable pairs exist only for $\mathbb{C}^d$19 or $\mathbb{C}^d$20: writing $\mathbb{C}^d$21 in the eigenbasis of $\mathbb{C}^d$22 forces either equal eigenspace multiplicities (hence trace zero, so $\mathbb{C}^d$23) or block-diagonal $\mathbb{C}^d$24 proportional to $\mathbb{C}^d$25, which is unimodular only when $\mathbb{C}^d$26 and corresponds precisely to Hermitian Hadamard matrices with constant diagonal. Thus the multiplying construction cannot be extended to generic signature parameters. One-parameter amicable families are known only for $\mathbb{C}^d$27 and $\mathbb{C}^d$28; the analogous families of orders 10 and 14 from Et-Taoui's work do not appear to yield amicable pairs. It also remains open whether the order-6 construction at $\mathbb{C}^d$29 is inequivalent to the general $\mathbb{C}^d$30 construction at $\mathbb{C}^d$31 (numerical evidence suggests it is). Finally, whether the multiplying or power constructions can produce 2-circulant ETFs — the object of the medium version of the doubling conjecture — is unresolved, as is the conjecture itself for general $\mathbb{C}^d$32.

Conclusion

The paper substantially enlarges the supply of known $\mathbb{C}^d$33 ETFs by introducing ETF-amicable pairs as a flexible algebraic interface between complex Hadamard theory and frame theory. The framework subsumes the Fallon–Iverson doubling construction, connects to classical amicable Hadamard pairs, resolves two of the four missing explicit dimensions below 150 ($\mathbb{C}^d$34 and $\mathbb{C}^d$35), and transfers Turyn's power construction wholesale to the ETF setting. Its reach is bounded by the classification of amicable pairs to $\mathbb{C}^d$36 and by the availability of structured Hadamard and conference matrices, leaving the full doubling conjecture open.