CATALOG ...

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$M_{13C}^{(2)}$

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A set of three Butson type matrices found by R. Nicoară, A. LaClair, N. Geist and A. Wintenberg in May 2016 (UT, Knoxville).

• $M_{13A}^{(0)}$
• $M_{13B}^{(1)}$
• $M_{13C}^{(2)}$

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$${\rm LOG}\left(M_{13C}\right)=\frac{1}{3}\pi \left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet &5 &5 &5 &4 &3 &3 &2 &2 &2 &2 &\bullet &\bullet\\ \bullet &2 &4 &2 &\bullet &1 &4 &1 &2 &4 &4 &\bullet &4\\ \bullet &2 &\bullet &\bullet &3 &2 &1 &3 &5 &5 &3 &5 &3\\ \bullet &2 &1 &4 &1 &4 &4 &5 &3 &1 &1 &5 &3\\ \bullet &2 &4 &1 &3 &4 &4 &1 &5 &1 &5 &3 &1\\ \bullet &\bullet &4 &2 &3 &5 &\bullet &4 &2 &\bullet &2 &2 &4\\ \bullet &5 &2 &2 &3 &2 &\bullet &5 &3 &3 &5 &5 &1\\ \bullet &4 &4 &\bullet &\bullet &1 &2 &4 &4 &2 &\bullet &2 &3\\ \bullet &\bullet &2 &4 &\bullet &3 &\bullet &2 &\bullet &2 &4 &3 &4\\ \bullet &4 &\bullet &4 &2 &5 &2 &\bullet &1 &4 &4 &2 &2\\ \bullet &4 &2 &2 &\bullet &5 &2 &2 &4 &5 &2 &4 &\bullet\\ \bullet &2 &2 &4 &4 &1 &4 &4 &\bullet &4 &1 &2 &\bullet\\ \end{array}\right].$$

Defect is $d(M_{13C})=2$.

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Extension of $M_{13C}$ to a $2$-parametric (maximal) affine family:

$$R_{M_{13C}^{(2)}}(b, c)=\left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & c & b & \bullet & \bullet & \bullet & c & \bullet & \bullet & b & \bullet & \bullet\\ \bullet & \bullet & c & b & b & b & \bullet & c & b & b & b & b & b\\ \bullet & \bullet & c & b & \bullet & \bullet & \bullet & c & \bullet & \bullet & b & \bullet & \bullet\\ \bullet & \bullet & c & b & c & c & \bullet & c & c & c & b & c & c\\ \bullet & \bullet & c & b & b & b & \bullet & c & b & b & b & b & b\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & c & b & \bullet & \bullet & \bullet & c & \bullet & \bullet & b & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & c & b & c & c & \bullet & c & c & c & b & c & c\\ \end{array}\right]$$

so that, for $b,c\in[0,1)$ one has

$$ M_{13C} \mapsto M_{13C}^{(2)}(b,c) = M_{13C} \circ {\rm EXP}\left(2 \pi i R_{M_{13C}^{(2)}}(b,c)\right). $$