Definition
A quaternary Hadamard matrix, also called a Butson Hadamard matrix BH(n, 4), is an n × n matrix H with entries in {1, -1, i, -i} satisfying
H H* = n In,
where H* is the conjugate transpose of H.
Quaternary Hadamard matrices are conjectured to exist for all even orders. Table 11.2 of the survey chapter of J. Seberry and M. Yamada, Hadamard matrices, sequences, and block designs, lists the odd numbers q below 1000 for which no quaternary Hadamard matrices of order 2q are known. Our work finds 117 of these unknown orders (the status at q = 163 is ambiguous, but the claimed construction cannot apply there, since 2q - 1 = 325 is not a prime power, so it is treated as an unknown order).
First, we provide a brief history of some unknown orders that have been found since the original survey was published. The first of the unknown orders fell a year after the survey: in 1993 Dragomir Djokovic constructed good matrices of order 127 (Good matrices of orders 33, 35, and 127 exist, J. Combin. Math. Combin. Comput. 14 (1993), pages 145 to 152), which can be used to construct a quaternary Hadamard matrix of order 254. A smaller order left blank in the same table, 94 (q = 47), was settled in 2026 by Ferenc Szollosi, who modified a block construction of Kharaghani and Seberry so that four circulant matrices of odd order p with entries 1 and -1, only two of them symmetric, combine into a quaternary Hadamard matrix of order 2p (arXiv:2603.09572).
Two further orders, 442 (QH442.txt) and 1574 (QH1574.txt), come from the symmetric conference matrices of order q2(q + 2) + 1 found by R. Mathon (Symmetric conference matrices of order pq2 + 1, Canad. J. Math. 30 (1978), pages 321 to 331), which give conference matrices at 442 (q = 7) and 1574 (q = 11); taking C + iI turns each into a quaternary Hadamard matrix. The two files are included solely for completeness and nothing is claimed for them here.
Twelve further orders, 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964, were settled recently by the discovery of real Hadamard matrices of those orders (see this page); a real Hadamard matrix is in particular a quaternary one.
In 2026, in collaboration with Hadi Kharaghani, Ali Mohammadian and Behruz Tayfeh-Rezaie, we uploaded a preprint, A search for Hadamard matrices of Williamson type (arXiv:2605.08661), which used near-Williamson matrices to find five previously unknown orders: 118, 130, 134, 206 and 266.
Twenty-three of the unknown orders, 142, 154, 222, 238, 262, 286, 302, 310, 326, 342, 366, 370, 382, 426, 430, 494, 518, 534, 686, 742, 754, 814 and 1558, were found with constructions of the Szollosi type, with two of the four blocks symmetric.
Twenty-seven more orders, 214, 406, 454, 470, 490, 538, 566, 590, 610, 638, 782, 806, 890, 926, 1066, 1142, 1306, 1310, 1358, 1462, 1466, 1526, 1646, 1690, 1706, 1882 and 1886, were found via unpublished approaches.
Lastly, sixty-two of the listed orders, 322, 330, 414, 474, 526, 546, 582, 646, 666, 714, 766, 786, 790, 834, 850, 862, 874, 894, 906, 934, 966, 990, 994, 1002, 1006, 1102, 1126, 1134, 1170, 1174, 1206, 1266, 1270, 1330, 1342, 1366, 1422, 1426, 1486, 1506, 1510, 1534, 1542, 1582, 1626, 1630, 1650, 1654, 1662, 1666, 1674, 1746, 1750, 1774, 1786, 1822, 1830, 1846, 1870, 1926, 1962 and 1966, come from a construction utilizing Paley-type matrices.
Three further orders, 1018, 1714 and 1766, follow from Williamson-type matrices of orders 509, 857 and 883, whose existence was shown by D. Djokovic (arXiv:1008.2043); such matrices can be used to construct quaternary Hadamard matrices of twice their order. The three files are included solely for completeness and nothing is claimed for them here.
Aside from the five orders discussed above (442, 1574, 1018, 1714 and 1766), the matrices listed below are of orders for which no such matrix was previously recorded in the literature.
Each file is plain text: n lines of n
characters, with 1 = 1, - = -1, i = i,
j = -i.
With this classification, the smallest unknown order is now 334. Among the even orders below 2000 there are 115 for which no quaternary Hadamard matrix is known to the author. The seven below 500 are 334, 358, 418, 438, 446, 478 and 498, and the full list is
334, 358, 418, 438, 446, 478, 498, 502, 506, 550, 574, 598, 606, 622, 658, 670, 682, 690, 694, 718, 738, 826, 838, 886, 910, 950, 958, 970, 974, 982, 986, 1026, 1030, 1038, 1046, 1054, 1074, 1078, 1106, 1146, 1150, 1162, 1166, 1178, 1190, 1198, 1210, 1222, 1242, 1246, 1258, 1262, 1274, 1286, 1294, 1314, 1318, 1334, 1338, 1378, 1386, 1390, 1394, 1398, 1414, 1438, 1442, 1446, 1450, 1474, 1478, 1498, 1502, 1530, 1562, 1566, 1570, 1578, 1586, 1590, 1606, 1634, 1678, 1702, 1718, 1726, 1730, 1738, 1782, 1794, 1798, 1806, 1810, 1814, 1818, 1826, 1834, 1838, 1842, 1854, 1866, 1894, 1898, 1910, 1918, 1930, 1938, 1942, 1946, 1958, 1970, 1978, 1982, 1986 and 1990.
Computations were performed on Digital Research Alliance of Canada systems. Parts of the search and of this website were prepared with the assistance of Claude (Anthropic).