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Abstract We show the optimal coherence of $2d$ lines in $$\mathbb{C}^{d}$$ is given by the Welch bound whenever a skew Hadamard matrix of order $d+1$ exists. Our proof uses a variant of Hadamard matrix doubling that converts any equiangular tight frame of size $$\tfrac{d-1}{2} \times d$$ into another one of size $$d \times 2d$$. Among $$d \leq 160$$, this produces equiangular tight frames of new sizes when $d = 11$, $35$, $39$, $43$, $47$, $59$, $67$, $71$, $83$, $95$, $103$, $107$, $111$, $119$, $123$, $127$, $131$, $143$, $151$ and $155$.
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Free, publicly-accessible full text available March 26, 2026
