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Each order is given with an expression, in the notation of the Hadamard Matrix Explorer, that builds a Hadamard matrix of that order. Up to 4096 the expressions are the program's stored table; above 4096 they are what H(n) works out on demand by the same rules. Every expression was built and verified. Names take numbers, matrices or sets of sequences, and the three are kept apart. q is always a prime power.
| Name | Meaning |
|---|---|
| Matrices from a number | |
S(n) | Sylvester. n a power of 2. |
P1(n) | Paley I. n−1 = q ≡ 3 (mod 4). |
P2(n) | Paley II. n/2−1 = q ≡ 1 (mod 4). |
W(n) | Williamson. n is the block size and the order is 4n; a stored set, or Turyn's construction when 2n−1 = q ≡ 1 (mod 4). |
Y(n) | Yamada. n = 4(q+2), q ≡ 1 (mod 4), with a skew-Hadamard matrix of order (q+3)/2 (q ≡ 5 (mod 8): Paley's, or a skew Goethals–Seidel matrix G(SK(k)) or G(A(k)), doubled as needed) or a symmetric conference matrix of that order (q ≡ 1 (mod 8): Paley's, Mathon's of order 442, or Seberry and Whiteman's of order 46). |
MC(n) | A symmetric conference matrix of order n/2, doubled as in Paley II: Mathon's (442 or 1574, stored) or Seberry and Whiteman's (46 or 3646, computed). |
CHK(x,y) | Craigen–Holzmann–Kharaghani. Complex Golay pairs of lengths x ≤ y give four complementary sequences of length x+y; order 8(x+y). The lengths come from stored pairs of lengths 1, 2, 3, 5, 10, 11, 13 by doubling and products. |
CG(P1(n)) | Miyamoto's Theorem 7. A complex Hadamard matrix of order q−1, taken from Paley I of order n = 2q−2 (q ≡ 3 (mod 4), n−1 a prime power), gives order 8q = 4(n+2). |
DG(7,n) | Kamali–Kharaghani. A dihedral Golay pair of length 7 with a binary Golay pair of length n; order 8(7+n). |
SY(q) | Seberry–Yamada. Order 8q, q a prime ≡ 3 (mod 4), from W(m), m = (q−1)/2, or their symmetric Williamson-type matrices of order m = 2p+1 (p, 2p−1 primes). |
CY(n) | Kumar. A stored quaternary conference matrix, his complex Yamada–Kiyasu array, Turyn's doubling; order n. |
Goethals–Seidel: G(set) has order 4 times the length | |
T(t) | T-sequences: a stored set, or built from a Golay pair when t = 2g+1. |
G(T(t),W(m)) | T-sequences combined with Williamson sequences by Turyn's construction; order 4tm. |
G(T(t),WT(m)) | Stored Williamson-type matrices (good matrices, 127; near-Williamson, 35, 59, 65, 67 and 93) in the Baumert–Hall array from T(t) (Cooper–Wallis); order 4tm. |
A(n) | A-sequences, from supplementary difference sets. |
SK(n) | Skew-type sequences; G(SK(n)) is skew-Hadamard (SK(1723) is Spence's). Also used by Y(n). |
TM(t) | A T-matrix, which meets the periodic condition only; with W(m) only when gcd(t,m) = 1. |
B(n,k) | Bordered B-sequences, or a stored matrix over a group; order 4(n+k). |
| Combining matrices | |
K(a,b,...) | Kronecker product. The orders multiply. |
M(a) | Miyamoto. a of order q−1, q ≡ 1 (mod 4); order 4q. |
M2(X) | Miyamoto II. X a symmetric Hadamard matrix of order (q−1)/2, q ≡ 1 (mod 4); order 4q. G(T(t),M2(X)) puts its four Williamson-type matrices in a Baumert–Hall array from T(t); order 4tq. |
G(T(t),P1(n)) | Paley I of order n (n−1 a prime ≡ 3 (mod 8)) cut into sixteen circulant blocks, times T(t) with gcd(t,n/4) = 1 (Seberry–Yamada); order tn. |
G(T(t),M(X)) | Miyamoto's array in commuting blocks, times T(t). X is WT(w) (q = 4w+1) or PG(m), a stored periodic Golay pair (q = 2m+1), q prime; order 4tq. |
AG(a,b) | Agaian product. Orders 4m and 4n give 8mn. |
| Inputs of M2 | |
P1S(n) | Paley I in a symmetric form. |
SH(116) | The stored symmetric Hadamard matrix of order 116 (Di Matteo, Djoković, Kotsireas). |
C(r) | The Paley conference matrix of order r+1, r ≡ 1 (mod 4); then q = 2r+3. |
SBH(86) | The stored symmetric complex Hadamard matrix BH(86, 4), for q = 173. |
The first route found wins, in this order: Sylvester, Paley I, Paley II, Williamson, Goethals–Seidel (T alone; T with W; T with WT; A; SK; TM alone; TM with W; TM with WT; B), MC, Miyamoto, Miyamoto II, T with Paley I in blocks, T with Miyamoto in blocks, then a Kronecker product, which folds the powers of 2 into one Sylvester factor. Agaian's product, Yamada's, CHK and CG, then DG, SY and CY only fill orders nothing above them reaches, in that order, so adding one never changes a route.