## a catalog (since 2006)

### by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski

Comprehensive study of Butson type matrices of size $2\leqslant N\leqslant 21$ can be found in the Butson Home by P. Lampio and F. Szöllősi.

$F_2^{(0)}$ $= H_2 \supset$ $F_2$ $\in BH(2,2)$ generic and other possible defect values $\in \{0\}$

$F_3^{(0)}$ $=$ $F_3$ $\in BH(3,3)$$\{0\} H_4 \simeq F_2\otimes F_2\in BH(4,2) \{3\} F_4^{(1)} \supset F_4 \in BH(4,4) \{\underline{1}, 3\} F_5^{(0)} = F_5 \in BH(5,5)$$\{0\}$

$S_6^{(0)}$ $=$ $S_6$ $\in BH(6,3)$ $\{0\}$

$D_6^{(1)}$ $\supset$ $D_6$ $\in BH(6,4)$ $\{\underline{4}\}$

$F_6^{(2)}$ $\supset$ $F_6$ $\simeq F_2\otimes F_3\in BH(6,6),\quad \left(F_6^{(2)}\right)^{\rm T}$ $\{\underline{4}\}$

$M_6^{(1)}$ $\{\underline{4}\}$

$X_6^{(2)}$ $\{\underline{4}\}$

$K_6^{(2)}$ $\{\underline{4}\}$

$K_6^{(3)}$ $\{\underline{4}\}$

$G_6^{(4)}$ $\{\underline{4}\}$

$P_7^{(1)}$ $\supset$ $P_7$ $\in BH(7,6)$ $\{\underline{2},3\}$

$F_7^{(0)}$ $=$ $F_7$ $\in BH(7,7)$ $\{0\}$

$Q_7^{(0)}$ $\{0\}$

$D_8^{(4)}$ $\supset$ $H_8$ $\simeq H_2\otimes H_2\otimes H_2\simeq H_4\otimes H_2\in BH(8,2)$ $\{\underline{5}, 7, 9, 15, 21\}$

$S_8^{(4)}$ $\supset$ $S_8$ $\in BH(8,4)$ $\{\underline{5},7,9,11,15,21\}$

$F_2\otimes F_4$ $\in BH(8,4)$$\{13\} F_8^{(5)} \supset F_8 \in BH(8,8) \{\underline{5}, 6, 7, 8, 9, 10, 11, 13\} A_8^{(0)} \{0\} T_8^{(1)} \supset \{ B_{8A}, B_{8B} \} \subset BH(8,20) \{\underline{3},7,11\} F_3\otimes F_3 \in BH(9,3)$$\{16\}$

$S_9^{(0)}$ $=$ $S_9$ $\in BH(9,6)$$\{0\} F_9^{(4)} \supset F_9 \in BH(9,9) \{\underline{4}, 6, 8, 10\} B_9^{(0)} = B_9 \in BH(9,10)$$\{2\}$

$N_9^{(0)}$ $\{0\}$

$Y_9^{(0)}$ $\{0\}$

$D_{10}^{(3)}$ $\supset$ $D_{10}$ $\in BH(10,4)$ $\{\underline{9},10,12,16\}$

$X_{10}^{(0)}$ $\in BH(10,5)$$\{0\} S_{10}^{(0)} = S_{10} \in BH(10,5) \{0\} D_{10\Sigma \ : \ \Sigma\in\{A,B\}}^{(7)} \supset \{ D_{10A}, D_{10B} \} \subset BH(10,6) \{\underline{8}_{A,B}, ...\} F_{10}^{(4)} \supset F_{10} \simeq F_2\otimes F_5\in BH(10,10),\quad \left(F_{10}^{(4)}\right)^{\rm T} \{\underline{8}\} N_{10A} \{0\} G_{10}^{(1)} \{8\} F_{11}^{(0)} = F_{11} \in BH(11,11) \{0\} N_{11A}^{(0)} \{0\} N_{11B}^{(0)} \{0\} H_{12}^{(7)} \supset H_{12} \in BH(12,2) \{[8, 31], [35, 38], 41, 42, 45, 49, 54, \underline{55}\} X_{12} \in BH(12,3)$$\{12\}$

$D_6\otimes F_2$ $\in BH(12,4)$$\{27\} D_{12} \in BH(12,4) \{45\} F_3\otimes F_2\otimes F_2 \simeq F_6\otimes F_2 \in BH(12,6)$$\{27\}$

$S_6\otimes F_2$ $\in BH(12,6)$$\{19\} F_{12\Sigma \ : \ \Sigma\in\{A,B,C,D\}}^{(9)} \supset F_{12} \simeq F_3\otimes F_4\in BH(12,12),\quad \left(F_{12\Sigma \ : \ \Sigma\in\{A,B,C,D\}}^{(9)}\right)^{\rm T} \{\underline{13}_{B,C,D}, 14, 15, 16, \underline{17}_A, 18, 19, 20, 21, 23\} S_{12}^{(5)} \supset S_{12} \in BH(12,36) \{\underline{11}, 13, 15, 17, 21, 27\} D_{12\Sigma \ : \ \Sigma\in\{A,B,C,...,R\}}^{(9)} \{...\} M_{13A} \in BH(13,6)$$\{0\}$

$M_{13B}$ $\in BH(13,6)$ $\{\underline{1}, 4\}$

$M_{13C}$ $\in BH(13,6)$ $\{\underline{2}, 3, 4, 6\}$

$F_{13}^{(0)}$ $=$ $F_{13}$ $\in BH(13,13)$ $\{0\}$

$P_{13(A)}^{(4)}$ $\supset$ $P_{13}$ $\in BH(13,60)$ $\{\underline{4},6\}$

$D_{14}^{(5)}$ $\supset$ $D_{14}$ $\in BH(14,4)$ $\{\underline{14},15,16,18,19,20,26,36\}$

$L_{14\Sigma \ : \ \Sigma\in\{A,B,...,N\}}^{(0)}$ $\in BH(14,4)$ $\{0\}$

$P_7\otimes F_2$ $\in BH(14,6)$$\{22\} S_{14}^{(0)} = S_{14} \in BH(14,7) \{0\} L_{14}^{(2)} \in BH(14,10)$$\{10\}$

$F_{14}^{(6)}$ $\supset$ $F_{14}$ $\simeq F_7\otimes F_2\in BH(14,14),\quad \left(F_{14}^{(6)}\right)^{\rm T}$$\{12\} FP_{14}^{(7)} , ...$$\{...\}$

$F_{15}^{(8)}$ $\supset$ $F_{15}$ $\simeq F_3\otimes F_5\in BH(15,15),\quad \left(F_{15}^{(8)}\right)^{\rm T}$ $\{\underline{16}\}$

$A_{15\Sigma \ : \ \Sigma\in\{A,B,C,...,H\}}$ $\{0,10,16\}$

$H_{16A}$ $\simeq H_2\otimes H_2\otimes H_2\otimes H_2\in BH(16,2)$$\{105\} H_{16B} \in BH(16,2)$$\{105\}$

$H_{16C}$ $\in BH(16,2)$$\{105\} H_{16D} \in BH(16,2)$$\{105\}$

$H_{16E}$ $\in BH(16,2)$$\{105\} F_4\otimes F_4 \in BH(16,4)$$\{57\}$

$S_8\otimes H_2$ $\in BH(16,4)$$\{39\} F_8\otimes H_2 \in BH(16,8)$$\{41\}$

$S_{16}^{(11)}$ $\supset$ $S_{16}$ $\in BH(16,8)$ $\{\underline{12},13,14,15,16,17,18,19,20, 23,24,27,32,39\}$

$L_{21}^{(0)}$ $\in BH(21,3)$$\{0\}$