New Hadamard matrices

Explicit Hadamard, skew-Hadamard, symmetric Hadamard and complex Hadamard matrices of orders for which no construction was published, or none could be traced; some were found concurrently with other groups (labels in NOVELTY.md). Every matrix has a certificate from which verify/ rebuilds it and checks it in exact arithmetic.

A Hadamard matrix of order n is an n × n matrix H with entries ±1 and H Hᵀ = n I. It can exist only for n = 1, 2 or n ≡ 0 (mod 4), and the Hadamard conjecture asserts that it exists for every multiple of 4. In August 2026 Alpöge, Voinov and Reynolds-Haertle constructed the twelve unresolved orders below 2000 [1], and Schneider constructed 2060 [2].

This repository constructs the next three, 2092, 2148 and 2284, and 31 more of the orders that Table 4 of Cati and Pasechnik [3] lists as unresolved, up to 11804; seven of the 34 were found concurrently by other groups (Unresolved orders). The smallest unresolved order is now 2396 = 4 · 599. On the basis of Table 4, 83 orders below 10000 (120 up to 4 · 2999) remain open, counting 2292 and 2588 (Middletownbooks/Hadamard [16]) as settled; 85 and 122 if they are not counted (FRONTIER.md).

Of the 95 orders up to 10000 that the survey of Egan, Kotsireas, London and Ó Catháin [18] lists as open, this repository constructs seven (2572, 3356, 3436, 3820, 4204, 5912 and 6796) and shows three to be known (6712, 8248 and 8696).

Most of the larger orders come from two block decompositions. Paley's matrix of order q + 1, q ≡ 3 (mod 8), is an array of sixteen circulant blocks [14], and Seberry and Yamada multiply such arrays by T-matrices [15]; we found the two results combined nowhere (hadamard/PALEY_BLOCKS.md). Miyamoto's matrix of order 4q [8] is an array of sixteen bordered blocks. When its input is a negaperiodic or periodic Golay pair, or symmetric Williamson-type matrices, and Paley's points are listed along a primitive root, these blocks commute and one permutation transposes them all, so T-matrices multiply this array as well; we found this stated nowhere (hadamard/MIYAMOTO_BLOCKS.md). For skew-Hadamard matrices the blocks of Paley's matrix give a theorem: a skew-Hadamard matrix of order 3(q + 1) exists for every prime power q ≡ 3 (mod 4) (skew/PALEY_SKEW.md).

Results

Family	Orders	New	New, but obtainable classically	First explicit construction	Concurrent	Known
Hadamard, unresolved in Table 4 [3]	34	11	14	2	7
Hadamard, entries of Table 4 lowered	3	3
Hadamard, recorded only through Seberry–Yamada's Corollary 8.8(i)	11			11
Hadamard, beyond Table 4 (12052 to 51580)	207	34	124	49
Hadamard, independent matrices of known orders	8					8
Skew-Hadamard (476 to 14384)	122	49	67	3		3
Symmetric Hadamard (260 to 9892)	103	36	49		9	9
Complex Hadamard (142 to 3398)	98	8		2	12	76
T-matrices (97, 109, 197)	3	3

The labels are defined in NOVELTY.md, which also lists the literature searched. "Obtainable classically" means that every ingredient, theorem or object, was published before 2026 and only their combination is new; a construction that needs a proof found in none of the sources read is New even when its inputs are classical. Known marks independent matrices of orders constructed before.

An entry n(m) of Table 4 gives the least m for which a Hadamard matrix of order 2ᵐn was known; the three lowered entries now have one of order 8n. The Hadamard orders beyond Table 4 (n > 2999) lie above the range of Cati–Pasechnik's table, and 40244, 45044 and 51580 also above the Handbook's Table 1.53; the complex orders 2102 and 3398 lie above Table 11.2 of the 1992 survey. These carry the labels "New, beyond the tables" and "New, beyond the tables, but obtainable classically", counted above under New and New, but obtainable classically.

The repository also holds the first explicit amicable Hadamard matrices of orders 52, 76, 100 and 124 and new orthogonal designs (skew), and Williamson-type matrices of order 111 (near-williamson).

Verification

python3 verify/verify.py                  # every real matrix and auxiliary object
python3 verify/verify.py 2092 skew/820    # selected orders
python3 verify/verify_complex.py          # the complex Hadamard matrices

verify.py rebuilds each matrix from its certificate.json, checks the defining identities of the ingredients (complementary sequences, T-matrices, Williamson-type matrices, conference matrices, skew or symmetric inputs), checks H Hᵀ = n I exactly, checks H + Hᵀ = 2I or H = Hᵀ where claimed, and compares the SHA-256 digest of the canonical CSV encoding with the recorded one. It needs Python 3 and NumPy. The orders below 12000 take a few minutes; the large orders dominate the running time, and 51580 takes about 40 minutes and 3.5 GB of memory. verify_complex.py rebuilds each quaternary matrix and checks every entry and K K* = n I exactly.

An independent checker in C reads a stored matrix and compares every pair of rows:

cc -O2 verify/check_matrix.c -o check_matrix      # add -fopenmp where the compiler supports it
gzip -dc hadamard/2092/H2092.csv.gz | ./check_matrix

Unresolved orders

For each order 4n, n is listed as unresolved in Table 4 of Cati and Pasechnik [3].

Order	n	Construction	Status
2092	523	Goethals–Seidel array [6] over ℤ₅₂₃, multiplier subgroup of order 9	Concurrent [16]
2148	537 = 3 · 179	bordered Goethals–Seidel array over ℤ₃ × ℤ₁₇₈, border of size 12	New
2284	571	Goethals–Seidel array over ℤ₅₇₁, three blocks related by a multiplier of order 3	Concurrent [13]
2572	643	quaternary conference matrix of order 322 (zero-diagonal Szöllősi array over ℤ₁₆₁ = F₇ × ℤ₂₃, a Paley plug-in, multiplier subgroup of order 3), complex Yamada–Kiyasu template (q = 641), Turyn's doubling	New
2636	659	bordered Goethals–Seidel array over ℤ₄ × ℤ₄ × ℤ₄₁, border of size 12	New
3156	789 = 3 · 263	Paley's matrix of order 1052 in sixteen circulant blocks [14], T-matrices of order 3 [15]	New, but obtainable classically
3356	839	bordered Goethals–Seidel array over ℤ₄ × ℤ₄ × ℤ₄ × ℤ₁₃ (Galois ring GR(4, 3) × ℤ₁₃), subgroup of index 8, border of size 28	New
3436	859	quaternary conference matrix of order 430 (zero-diagonal Szöllősi array over ℤ₂₁₅, multiplier subgroup of order 6), complex Yamada–Kiyasu template (q = 857), Turyn's doubling	New
3732	933 = 3 · 311	Goethals–Seidel array over ℤ₃ × GF(311), multiplier subgroup of order 31; skew	Concurrent [16, 18]; skew: New
3820	955 = 5 · 191	quaternary conference matrix of order 478 (zero-diagonal Szöllősi array over ℤ₂³ × ℤ₂₉ with a border of the seven nontrivial characters of ℤ₂³, symmetry group of order 7), complex Yamada–Kiyasu template (q = 953), Turyn's doubling	New
3964	991	bordered Goethals–Seidel array over ℤ₂ × ℤ₂ × ℤ₂₄₇, border of size 12	New
4204	1051	quaternary conference matrix of order 526 (bordered zero-diagonal Szöllősi array over ℤ₂₆₂, multiplier subgroup of order 5), complex Yamada–Kiyasu template (q = 1049), Turyn's doubling	New
4492	1123	Goethals–Seidel array over ℤ₁₁₂₃, multiplier subgroup of order 33; a skew matrix of this order is in skew/4492	Concurrent [18]; skew: New
5020	1255 = 5 · 251	Goethals–Seidel array over ℤ₅ × GF(251), multiplier subgroup of order 50	Concurrent [18]
5260	1315 = 5 · 263	Paley's matrix of order 1052 in sixteen circulant blocks, T-matrices of order 5	New, but obtainable classically
5932	1483	Goethals–Seidel array over ℤ₁₄₈₃, multiplier subgroup of order 39	Concurrent [18]
6036	1509 = 3 · 503	Paley's matrix of order 2012 in sixteen circulant blocks, T-matrices of order 3	New, but obtainable classically
6108	1527 = 3 · 509	Miyamoto's matrix [8] in sixteen bordered blocks with near-Williamson matrices of order 127 [11], T-matrices of order 3	First explicit construction
6356	1589 = 7 · 227	Paley's matrix of order 908 in sixteen circulant blocks, T-matrices of order 7	New, but obtainable classically
6756	1689 = 3 · 563	Paley's matrix of order 2252 in sixteen circulant blocks, T-matrices of order 3	New, but obtainable classically
6796	1699	quaternary conference matrix of order 850 (zero-diagonal Szöllősi array over ℤ₅ × ℤ₅ × ℤ₁₇, symmetry group of order 24), complex Yamada–Kiyasu template (q = 1697), Turyn's doubling	New
7364	1841 = 7 · 263	Paley's matrix of order 1052 in sixteen circulant blocks, T-matrices of order 7	New, but obtainable classically
7532	1883 = 7 · 269	Miyamoto's matrix in sixteen bordered blocks with near-Williamson matrices of order 67, T-matrices of order 7	New
7660	1915 = 5 · 383	Paley's matrix of order 1532 in sixteen circulant blocks, T-matrices of order 5	New, but obtainable classically
8620	2155 = 5 · 431	Paley's matrix of order 1724 in sixteen circulant blocks, T-matrices of order 5	New, but obtainable classically
9172	2293	Miyamoto's theorem [8] from the matrix of order 2292	Concurrent [18]
10060	2515 = 5 · 503	Paley's matrix of order 2012 in sixteen circulant blocks, T-matrices of order 5	New, but obtainable classically
10180	2545 = 5 · 509	Miyamoto's matrix in sixteen bordered blocks with near-Williamson matrices of order 127, T-matrices of order 5	First explicit construction
10644	2661 = 3 · 887	Paley's matrix of order 3548 in sixteen circulant blocks, T-matrices of order 3	New, but obtainable classically
10932	2733 = 3 · 911	Paley's matrix of order 3644 in sixteen circulant blocks, T-matrices of order 3	New, but obtainable classically
11260	2815 = 5 · 563	Paley's matrix of order 2252 in sixteen circulant blocks, T-matrices of order 5	New, but obtainable classically
11292	2823 = 3 · 941	Miyamoto's matrix in sixteen bordered blocks with symmetric Williamson-type matrices of order 235 from a symmetric conference matrix of order 118 found here, T-matrices of order 3	New
11572	2893 = 11 · 263	Paley's matrix of order 1052 in sixteen circulant blocks, T-matrices of order 11	New, but obtainable classically
11804	2951 = 13 · 227	Paley's matrix of order 908 in sixteen circulant blocks, T-matrices of order 13	New, but obtainable classically

2284 also follows from the quaternary Hadamard matrix of order 1142 that Zaitsev posted concurrently [13], by Turyn's doubling.

Priority, where other groups are involved:

• Our matrix of order 2292 = 4 · 573 (Spence's construction over ℤ₂ × ℤ₂ × ℤ₁₄₃) is not in the table. Middletownbooks/Hadamard had published an independent matrix of that order on 9 October 2026, before ours of 14 October, and we learned of it on 19 October [16] (hadamard/2292).
• The same repository settled 2588 = 4 · 647 on 14 October 2026 and committed independent matrices of orders 2092 and 3732 that day. Ours were committed on 11 October (2092, 4492, 5932) and 13 October (3732, 5020).
• The survey of Egan, Kotsireas, London and Ó Catháin (19 October 2026) constructs 3732, 4492, 5020 and 5932 with the multiplier subgroups of orders 31, 33, 50 and 39 that ours use, credits Cooley with 2092, and credits this repository with 2148, 2284, 2636 and 3964 [18]. It also derives 9172 from Cooley's matrix of order 2292 by Miyamoto's theorem; ours, from our own matrix of order 2292, dates from 14 October.
• We claim no priority for the seven Concurrent orders. Neither of the other matrices of order 3732 is stated to be skew, and the survey does not state that its matrix of order 4492 is skew; decoded from the images of the Hadamard Matrix Explorer on 2 November 2026, its matrices of orders 3732 and 4492 are not skew (NOVELTY.md). Our skew matrix of order 4492 is a separate one (skew/4492).
• Đoković claimed 1527 and 2545 in 2010 through Seberry–Yamada's Corollary 8.8(i), whose printed proof does not give Williamson-type matrices (NOTES.md §7).

Other results

• Hadamard matrices beyond Table 4: 207 orders from 12052 to 51580. 158 of them are listed as unknown in Table 1.53 of the Handbook or lie above its range, and 49 are recorded as known without a valid or traceable construction. Most come from the two block decompositions, the rest from the recursive theorems of Miyamoto [8], Yamada–Kiyasu [9] and Cooper–Wallis [7] applied to matrices here and to objects published in 2026 (hadamard).

• Entries of Table 4 lowered. Three entries n(m) of Table 4 with m ≥ 4 now have a Hadamard matrix of order 8n:

  • 5912 = 8 · 739 (Table 4: 739(6); 739(4) by Craigen–Holzmann–Kharaghani's rule, which the table omits; one of the six open orders 8p of [18]), from Spence's bordered construction over the group ℤ₂₁₁ × F₇ of odd order 1477 (hadamard/5912). Two orders 8p remain open, 4792 and 5752.
  • 12472 = 8 · 1559 (Table 4: 1559(4)), from Miyamoto's Theorem 7 written out for the quaternary Hadamard matrix of order 1558 of 2026. On the way it gives a quaternary Hadamard matrix of order 6236 = 4 · 1559, an order with no known real one (hadamard/COMPLEX_MIYAMOTO.md).
  • 20712 = 8 · 3 · 863 (Table 4: 2589(4)), from Miyamoto's quaternary array in sixteen commuting blocks, doubled, times T-matrices (hadamard/QUATERNARY_MIYAMOTO_BLOCKS.md).

• Corollary 8.8(i). 68 orders were claimed or recorded as known only through Seberry–Yamada's Corollary 8.8(i): ten that Table 4 omits, 6108, 10180 and 10284, 12952 (Table 4's entry 1619(3)), and 54 beyond Table 4. We give the first valid constructions of 59: the ten that Table 4 omits (2076 to 10444), 6108, 10180 and 12952, and 46 beyond Table 4 (hadamard). The other nine (10284, 15612, 17140, 17916, 23996, 27252, 33308, 36428 and 37708, that is n = 2571, 3903, 4285, 4479, 5999, 6813, 8327, 9107 and 9427) have no valid construction known to us (NOTES.md §7). The corollary's matrices still give a complex Hadamard matrix of order 2q from any quaternary Hadamard matrix of order (q − 1)/2 (NOTES.md §19).

• Skew-Hadamard matrices of 119 orders: Goethals–Seidel difference families with a skew block found by search, Goethals–Seidel arrays with negacyclic blocks searched over ℤ[i] (orders 8v: 1528, 2216, 2392 and 4712), Seberry's theorem with new amicable pairs and orthogonal designs, Wallis's product, and the 3(q + 1) theorem, which settles 52 values v < 2500 that the Đoković–Seberry census [4] lists as undecided (skew). The smallest of these orders is 476 (v = 119, the census's fourth smallest undecided value), from a Paley product with many w-positions; 980 (v = 245), 1540 (v = 385), 2156 (v = 539) and 3124 (v = 781) come from the same structure with a multiplier on K. 1004 (v = 251), 2212 (v = 553), 3044 (v = 761), 3364 (v = 841) and 4492 (v = 1123), also undecided in the census, come from low-energy k-trees with multiplier groups of orders 5, 13, 19, 21 and 33; for 1004 the lists are generated by Metropolis chains, and for 3044 the reversal twins are removed from them. 3364 = 4 · 29² is a family over ℤ₂₉ × ℤ₂₉ = GF(841)⁺; the smallest order with odd part 841 known before was 8 · 841 = 6728 (Momihara and Xiang; Cati–Pasechnik's Table 5 prints 2⁶ · 841). 2300 (v = 575) is a Paley product found by the same k-tree, and 6628 (v = 1657) and 7204 (v = 1801) come from Đoković's spin structure, found by a tabu search. With these, 107 of the 737 values that the census lists as undecided for skew matrices are decided and 630 are not.

• Symmetric Hadamard matrices of 94 orders that the census lists as undecided:

  • 260 (v = 65, the census's smallest value), from Verma's quaternary Legendre pair of length 64 (2026);
  • 532, 732, 876, 4828 and 5700, from searches and plug-in constructions;
  • 48 orders 4q (q ≡ 1 mod 8 a prime power), from Seberry–Yamada's Williamson-type matrices with a repeated block in Seberry and Balonin's propus array, among them 7012 = 4 · 1753 from our matrix of order 876 (symmetric/PROPUS_MIYAMOTO.md);
  • 39, from three constructions that need no symmetric input:
    • order 4q from any Hadamard matrix of order (q − 1)/2, q ≡ 1 (mod 8) (symmetric/PROPUS_WEIGHING.md, 10 orders);
    • order 4(q + 2) from any skew-Hadamard matrix of order (q + 3)/2, q ≡ 5 (mod 8), Yamada's matrix in symmetric quaternary form (symmetric/YAMADA_KIYASU.md, 26 orders, 8 of them Concurrent because quaternary matrices on Zaitsev's page [13] are equivalent to symmetric ones);
    • order 4q, q ≡ 5 (mod 8), from a Hadamard matrix [[P, Q], [Q, P]] of order q − 1 (symmetric/PERIODIC_GOLAY.md, 404, 596 and 1556);
  • 940 (v = 235), from a symmetric conference matrix of order 118 found by an exact cover, through Seberry–Balonin's Corollary 1. It is Concurrent with the conference graph on 117 vertices that J. Y. Z. Ho reported in October 2026 (symmetric/940). The same matrix gives 11292 = 4 · 2823 in the table above.

  Not among the 94 are nine Known census values that follow from published theorems the census omits: 1804 and 3532 from Seberry–Balonin's Corollary 1, and 612, 1300, 1740, 2812, 4732, 4900 and 7564 from Spence's Corollary 3 (1977), rebuilt here by other constructions (symmetric/CONFERENCE_PLUG.md). With the three that Đoković settled (arXiv:2508.17141), 106 of the 595 values that the census lists as undecided for symmetric matrices are decided and 489 are not.

• Complex Hadamard matrices (complex-hadamard):

  • the 12 orders 214, 238, 302, 326, 366, 382, 426, 494, 518, 534, 754 and 1558, blank in Seberry and Yamada's table [5] and in the papers since [10, 11], from searches over Szöllősi's array [10] and its bordered, character-bordered and twisted forms. Zaitsev found all twelve concurrently and independently [13]; we claim no priority;
  • the quaternary matrices of orders 358, 418 and 498, open on Zaitsev's list: 418 from a Paley plug-in in Szöllősi's array over ℤ₁₉ × F₁₁, and 358 and 498 from the character-bordered array with three characters over ℤ₂⁴ × ℤ₁₁ and ℤ₂ × ℤ₃ × ℤ₄₁;
  • quaternary conference matrices of orders 322, 430, 478, 526 and 850, which give quaternary Hadamard matrices of orders 1286, 1718 and 1910 (open on Zaitsev's list), 2102 and 3398 and, by Turyn's doubling, the Hadamard matrices of orders 2572, 3436, 3820, 4204 and 6796. The smallest quaternary order that remains open is 334;
  • the first explicit matrices of 1618 and 1906, and explicit matrices of 76 orders 2N with 4N − 1 a prime power, which Ito's dicyclic Hadamard groups and Egan's Theorem 4.5 [12] give.

• T-matrices of orders 97, 109 and 197 (t-matrices). 97 and 109 have the form 6m + 1 and may appear in Lou, Xia and Xia's work on T-matrices of that order, which we could not read.

• Periodic Golay pairs of length 194 (one class, with a multiplier group of order 3 over ℤ₂ × ℤ₉₇) and of length 146 (four classes, over ℤ₂ × ℤ₇₃), both on Đoković and Kotsireas's list of undecided lengths below 300, from an exhaustive row enumeration (search/pgp, search/NOTES.md §113). The pair of length 194 gives symmetric Hadamard matrices of order 1556 above. These are search results with certificates for the matrix of order 1556 only.

• Classifications by exhaustive search with the fibre-completion engines of FLC [17]: good matrices of order 55 form exactly one class, and there are no Williamson matrices and no good matrices of order 65 (search/NOTES.md §96, §101). Both orders 65 had been listed as open. These are search results, not certificates.

Methods

Construction	Orders	Details
Goethals–Seidel difference families invariant under a multiplier subgroup, found by an extended k-tree search	2092, 3532, 4492, 5932; skew 4492	METHOD.md, search/NOTES.md §110
The same with blocks permuted by a multiplier of order 3	2284	METHOD.md
Low-energy k-tree for skew Goethals–Seidel families over ℤ_p, with lists generated by Metropolis chains (1004) or with reversal twins removed (3044)	skew 1004, 3044	search/listgen, search/NOTES.md §111
Low-energy k-tree for a skew Goethals–Seidel family over ℤ₂₉ × ℤ₂₉ = GF(841)⁺ with a multiplier group of order 21 (index 40, no factor ℤ_m)	skew 3364	skew/3364, search/NOTES.md §112
Tabu search for skew Goethals–Seidel families with Đoković's spin structure, on the centred hexagonal primes 1657 and 1801	skew 6628, 7204	skew/6628, search/NOTES.md §33, §115
Bordered Goethals–Seidel arrays with exact compression chains (index 4, and index 8 over a Galois ring for 3356)	2148, 2636, 3356, 3964	METHOD.md, search/NOTES.md §18–19, §102, search/bgschain
Cyclotomic products ℤ_m × GF(q) of index 4 to 10	3732, 5020; skew 820, 1044, 1204, 1276, 2212, 3380, 5668	search/NOTES.md §16, §20, §110
Spence's construction with blocks (A, B, B∘μ, B∘μ²) and a Legendre lift	2292	NOTES.md §13, search/spence_spin
Spence's construction over a group of odd order 2p − 1, a Paley plug-in with a multiplier group of order 45, calibrated by near misses	5912	hadamard/5912, search/eightp, search/NOTES.md §108
Paley's matrix in sixteen circulant blocks, times T-matrices	157 orders, 14 of them unresolved	hadamard/PALEY_BLOCKS.md
Miyamoto's matrix in sixteen commuting bordered blocks, times T-matrices	50 orders, 4 of them unresolved	hadamard/MIYAMOTO_BLOCKS.md
Miyamoto's quaternary array in sixteen commuting blocks, doubled, times T-matrices	20712	hadamard/QUATERNARY_MIYAMOTO_BLOCKS.md
Recursive theorems of Miyamoto (Theorem 7 in quaternary form for 12472), Yamada–Kiyasu and Cooper–Wallis	9172, 12472 and orders beyond Table 4	hadamard, williamson-type, t-matrices
Skew difference families: compression chains, Paley products, low-energy k-trees	skew orders	skew, search/NOTES.md §10–16, §20, §87–90, §110–111
Goethals–Seidel arrays with negacyclic blocks of order 2v, as quaternary sequences on ℤ_v: multiplier groups and Paley plug-ins	skew 1528, 2216, 2392, 4712	skew/NEGACYCLIC.md, search/negaskew, search/NOTES.md §88–89
Paley's matrix in sixteen negacyclic blocks, Frobenius pairing, T-matrices of order 3	skew orders 3(q + 1)	skew/PALEY_SKEW.md
Propus quadruples, the symmetric m(m − 1) and m(m + 3) constructions, Miyamoto's lemma in the propus array	symmetric orders	symmetric, search/NOTES.md §17, §27–31
A Hadamard matrix of order (q − 1)/2 in the propus array; Yamada–Kiyasu in symmetric quaternary form	symmetric orders	symmetric, NOTES.md §30–31
A symmetric conference matrix of order 118 found by an exact cover, in Seberry–Balonin's Corollary 1 and in Miyamoto's blocks	symmetric 940; 11292	symmetric/940, search/NOTES.md §95, search/conference118
[[P, Q], [Q, P]] with Paley's matrices from a periodic Golay pair; a symmetric conference matrix in I ⊗ X + C ⊗ Y (the case r = 1 of Spence's theorem)	symmetric 404, 596, 1556; 612, 1300, 1740, 4900, 7564 (Known)	symmetric/PERIODIC_GOLAY.md, symmetric/CONFERENCE_PLUG.md, search/NOTES.md §65, §68, §113
Exhaustive row enumeration of periodic Golay pairs over ℤ₂ × ℤ_w invariant under a multiplier group of odd order (ternary rows constant on the orbits, parity-key buckets, support join)	the pair of length 194 behind symmetric 1556; four pairs of length 146	search/pgp, search/NOTES.md §113
Szöllősi's array and its bordered, character-bordered and twisted forms	complex orders	complex-hadamard
Quaternary conference matrices from the zero-diagonal Szöllősi array, plain, bordered or with a border of characters (for 2572, 3436 and 4204 found by two-level sampling: compress, sample compressed instances, complete each exactly through its fibres, as in FLC [17]; for 2572 on a Paley plug-in structure), in the complex Yamada–Kiyasu template (as in Zaitsev's QH470 to QH1526 [13]), and Turyn's doubling	2572, 3436, 3820, 4204, 6796	hadamard/6796, NOTES.md §31, §34, search/NOTES.md §74–83

NOTES.md and search/NOTES.md are the research notebooks: the constructions, the structures that turned out to be empty, the obstructions, and calibrations of the solution-count estimates. FRONTIER.md lists the unresolved orders up to 4 · 2999 with the cheapest search structures known for each.

Layout

Path	Contents
hadamard/<order>/, skew/<order>/, symmetric/<order>/	certificate.json, README.md, and where stored sequences.txt and the matrix H<order>.csv.gz (one row per line, entries separated by commas)
complex-hadamard/	certificates C<order>.json and the constructions
t-matrices/, near-williamson/, williamson-type/	auxiliary objects
verify/	the verifiers
search/	the search programs, with search/README.md as an index
NOVELTY.md	status labels, reference tables, the literature searched for each order, corrections
FRONTIER.md	the unresolved orders up to 4 · 2999 and the cheapest search structures for each
METHOD.md	the Goethals–Seidel search: search space, algorithm, cost model, block symmetries, bordered structures
NOTES.md, search/NOTES.md	the structural and the search notebooks
CITATION.cff, LICENSE	citation metadata, Apache License 2.0

References

1. L. Alpöge, P. Voinov, S. Reynolds-Haertle, Hadamard matrices of orders 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964, announcement, 12 August 2026.
2. Schneider, Hadamard matrix of order 2060, public gist, 23 August 2026.
3. M. Cati, D. V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897v2 (2025).
4. D. Ž. Đoković, J. Seberry, census of skew and symmetric Hadamard matrices of order 4v for odd v < 2500, 28 May 2025.
5. J. Seberry, M. Yamada, Hadamard matrices, sequences, and block designs, in J. H. Dinitz, D. R. Stinson (eds.), Contemporary Design Theory: A Collection of Surveys, Wiley (1992) 431–560.
6. J.-M. Goethals, J. J. Seidel, A skew Hadamard matrix of order 36, J. Austral. Math. Soc. 11 (1970) 343–344.
7. J. Cooper, J. Wallis, A construction for Hadamard arrays, Bull. Austral. Math. Soc. 7 (1972) 269–278.
8. M. Miyamoto, A construction of Hadamard matrices, J. Combin. Theory Ser. A 57 (1991) 86–108.
9. M. Yamada, Some new series of Hadamard matrices, J. Austral. Math. Soc. Ser. A 46 (1989) 371–383.
10. F. Szöllősi, A complex Hadamard matrix of order 94, arXiv:2603.09572 (2026).
11. H. Kharaghani, A. Mohammadian, B. Tayfeh-Rezaie, V. Zaitsev, A search for Hadamard matrices of near-Williamson type, arXiv:2605.08661v2 (6 September 2026). Title and authors as in the PDF of v2; the arXiv listing shows the title of v1 and three authors.
12. R. Egan, Phased unitary Golay pairs, Butson Hadamard matrices and a conjecture of Ito's, Des. Codes Cryptogr. 87 (2019) 67–74.
13. V. Zaitsev, Quaternary Hadamard matrices, https://www.cs.uleth.ca/~zaitsevv/quaternary.html, last updated 14 October 2026.
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16. Middletownbooks, Hadamard (GitHub repository), https://github.com/Middletownbooks/Hadamard (order 2292 committed 9 October 2026; orders 2092, 2588 and 3732 committed 14 October 2026).
17. Kishan, FLC: exact classification of circulant Williamson, good and best matrices by fibre-lattice completion (unpublished software, 2026).
18. R. Egan, I. Kotsireas, S. London, P. Ó Catháin, Recent progress in the construction of Hadamard matrices, arXiv:2610.27175 (19 October 2026).

Citation

If you use these matrices or programs, please cite the repository (CITATION.cff):

Kishan, New Hadamard matrices, https://github.com/vatskishan03/Hadamard, 2026.

License

Apache License 2.0; see LICENSE.