Complex Hadamard Matrices

a catalogue (since 2006)

by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski

old version

including Butson-type matrices

Butson type matrices are listed in a dephased, log-Hadamard form.

For any representant of an affine Hadamard family \(H(a)=F\circ{\rm EXP}(i a)\) being a Butson type \(H(a)\in BH(N, q)\) there exists another parameter \(b=2\pi / k q:k\in\mathbb{Z}\) such that \(H(b)\in BH(N, qk)\). Due to this fact the set of all Butsons is countable but not finite. Thus from an affine family of complex Hadamard matrices we provide only the Butson representant with the minimal factor \(k\).

Comprehensive study of Butson type matrices of size \(N\leqslant 21\) 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 defect values = \(\{0\}\)

\(F_3^{(0)}\) \(=\) \(F_3\) \(\in BH(3,3)\)\(\{0\}\)

\(H_4\) \(\simeq F_2\otimes F_2\in BH(4,2)\) \(\{3\}\)

\(F_4^{(1)}\) \(\supset\) \(F_4\) \(\in BH(4,4)\) \(\{1\}\)

\(F_5^{(0)}\) \(=\) \(F_5\) \(\in BH(5,5)\)\(\{0\}\)

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

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

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

\(B_6^{(1)}\) \(\supset\) \(C_6^{(0)}\) \(\{4\}\)

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

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

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

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

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

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

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

\(C_{7\Sigma}^{(0)}\) \(: \Sigma\in\{A,B,C,D\}\) \(\{0\}\)

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

\(D_8^{(4)}\) \(\supset\) \(H_8\) \(\simeq H_2\otimes H_2\otimes H_2\simeq H_4\otimes H_2\in BH(8,2)\)\(\{21\}\)

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

\(F_2\otimes F_4\) \(\in BH(8,4)\)\(\{13\}\)

\(F_8^{(5)}\) \(\supset\) \(F_8\) \(\in BH(8,8)\) \(\{5\}\)

\(V_{8\Sigma}^{(0)}\) \(: \Sigma\in\{A,B,C,D\}\) \(\{0\}\)

\(A_8^{(0)}\) \(\{0\}\)

\(T_8^{(1)}\) \(\supset \{\) \(B_{8A}\), \(B_{8B}\) \( \} \subset BH(8,20)\) \(\{3,7,11\}\)

\(F_3\otimes F_3\) \(\in BH(9,3)\)\(\{16\}\)

\(S_9^{(0)}\) \(=\) \(S_9\) \(\in BH(9,6)\)\(\{0\}\)

\(F_9^{(4)}\) \(\supset\) \(F_9\) \(\in BH(9,9)\) \(\{4\}\)

\(B_9^{(0)}\) \(=\) \(B_9\) \(\in BH(9,10)\)\(\{2\}\)

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

\(K_9^{(2)}\) \(\{2,4,10,12,16\}\)

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

\(D_{10\Sigma}^{(7)}\) \(: \Sigma\in\{A,B\}\) \(\{8,10,16\}\)

\(X_{10}^{(0)}\) \(\in BH(10,5)\)\(\{0\}\)

\(S_{10}^{(0)}\) \(=\) \(S_{10}\) \(\in BH(10,5)\) \(\{0\}\)

\(F_{10}^{(4)}\) \(\supset\) \(F_{10}\) \(\simeq F_2\otimes F_5\in BH(10,10),\left(F_{10}^{(4)}\right)^{\rm T}\)\(\{8\}\)

\(N_{10A}\) \(\{0\}\)

\(N_{10B}^{(3)}\) \(\{7\}\)

\(G_{10}^{(1)}\) \(\{8\}\)

\(F_{11}^{(0)}\) \(=\) \(F_{11}\) \(\in BH(11,11)\) \(\{0\}\)

\(C_{11\Sigma}^{(0)}\) \(: \Sigma\in\{A,B\}\) \(\{0\}\)

\(N_{11A}^{(0)}\) \(\{0\}\)

\(N_{11B}^{(0)}\) \(\{0\}\)

\(Q_{11\Sigma}^{(0)}\) \(: \Sigma\in\{A,B\}\) \(\{0\}\)

\(H_{12}^{(7)}\) \(\supset\) \(H_{12}\) \(\in BH(12,2)\)\(\{55, ...\}\)

\(X_{12}\) \(\in BH(12,3)\)\(\{12\}\)

\(D_6\otimes F_2\) \(\in BH(12,4)\)\(\{27\}\)

\(D_{12}\) \(\in BH(12,4)\) \(\{45\}\)

\(F_3\otimes F_2\otimes F_2\) \(\simeq F_6\otimes F_2\) \(\in BH(12,6)\)\(\{27\}\)

\(S_6\otimes F_2\) \(\in BH(12,6)\)\(\{19\}\)

\(L_{12}^{(0)}\) \(\in BH(12,6)\)\(\{0\}\)

\(\qquad ...\)

\(F_{12\Sigma}^{(9)}\) \(\supset\) \(F_{12}\) \(\simeq F_3\otimes F_4\in BH(12,12),\left(F_{12\Sigma}^{(9)}\right)^{\rm T} : \Sigma\in\{A,B,C,D\}\)\(\{17\}\)

\(S_{12}^{(5)}\) \(\supset\) \(S_{12}\) \(\in BH(12,36)\) \(\{17\}\)

\(D_{12\Sigma}^{(9)}\) \(: \Sigma\in\{A,B,...,R\}\) [103] \(\{...\}\)

\(M_{13A}\) \(\in BH(13,6)\)\(\{0\}\)

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

\(M_{13C}\) \(\in BH(13,6)\)\(\{2\}\)

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

\(P_{13}^{(4)}\) \(\supset\) \(P_{13}\) \(\in BH(13,60)\) \(\{4,5\}\)

\(C_{13\Sigma}^{(0)}\) \(: \Sigma\in\{A,B\}\) \(\{0\}\)

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

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

\(\qquad ...\)

\(P_7\otimes F_2\) \(\in BH(14,6)\)\(\{22\}\)

\(S_{14}^{(0)}\) \(=\) \(S_{14}\) \(\in BH(14,7)\) \(\{0\}\)

\(L_{14}^{(2)}\) \(\in BH(14,10)\)\(\{10\}\)

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

\(FP_{14}^{(7)}\) \(, ...\)\(\{...\}\)

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

\(A_{15\Sigma}\) \(\{0,10,16\}\)

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

\(H_{16B}\) \(\in BH(16,2)\)\(\{105\}\)

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

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

\(H_{16E}\) \(\in BH(16,2)\)\(\{105\}\)

\(L_{16}^{(0)}\) \(\in BH(16,4)\)\(\{0\}\)

\(\qquad ...\)

\(F_4\otimes F_4\) \(\in BH(16,4)\)\(\{57\}\)

\(S_8\otimes H_2\) \(\in BH(16,4)\)\(\{39\}\)

\(F_8\otimes H_2\) \(\in BH(16,8)\)\(\{41\}\)

\(S_{16}^{(11)}\) \(\supset\) \(S_{16}\) \(\in BH(16,8)\) \(\{25\}\)

\(F_{16}^{(17)}\) \(\supset\) \(F_{16}\) \(\in BH(16,16)\) \(\{17\}\)

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

\(\qquad ...\)