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I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power. Since $$ n!-1 \bmod 4 = 3 $$ construction 1 has to be chosen:

If $q$ is congruent to $3 (\bmod 4)$ [and $Q$ is the corresponding Jacobsthal matrix] then $$ H=I+\begin{bmatrix} 0 & j^T\\ -j & Q\end{bmatrix} $$ is a Hadamard matrix of size $q + 1$. Here $j$ is the all-1 column vector of length $q$ and $I$ is the $(q+1)×(q+1)$ identity matrix. The matrix $H$ is a skew Hadamard matrix, which means it satisfies $H+H^T = 2I$.

The problem is that the number of primes among $n!-1$ is restricted (see A002982). I checked the values of $n!-1$ given by Wolfram|Alpha w.r.t. be a prime power, without success, so Payley's construction won't work for all $n$.

Is there a general way to get the matrices, or is it case by case different?

I haven't yet looked into Williamson's construction nor Turyn type constructions. Would it be worth a closer look (sure it would, but) concerning my problem? Where can I find their constructions?

PS for the interested reader: I've found a nice compilation of Hadamard matrices here: http://neilsloane.com/hadamard/

mathma
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1 Answers1

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I don't think a general construction for Hadamard matrices of order $n!$ is known. The knowledge about general construction methods for Hadamard matrices is quite sparse, the basic ones (see also the Wikipedia article) are:

  1. If $n$ is a multiple of $4$ such that $n-1$ is a prime power or $n/2 - 1$ is a prime power $\equiv 1\pmod{4}$, then there exists a Hadamard Matrix of order $n$ (Paley).

  2. If $n$ is a multiple of $4$ such that there exists a Hadamard Matrix of order $n/2$, then there exists a Hadamard Matrix of order $n$ (Sylvester).

The Hadamard conjecture states that for all multiples $n$ of $4$ there is a Hadamard matrix of order $n$. The above constructions do not cover all these $n$, the smallest case not covered is $n = 92$. There are more specialized constructions and a few computer constructions, such that the smallest open case is $n = 668$ nowadays.

EDIT

I have just checked the range $n\in\{1,\ldots,100\}$. In all but $9$ cases, a combination of the above Paley and Sylvester constructions yields a Hadamard matrix of order $n!$. The exceptional cases are $n\in\{13,26,44,52,63,67,70,77,85\}$, where more specialized constructions like Williamsons' one are needed.

EDIT2

Upon request, I'm adding details on $n=88$. The construction is Paley and then 31 times Sylvester. To apply Paley, we need that $88!/2^{31} - 1$ is prime. This can easily be checked with in the online calculator of the Magma computer algebra system.

> IsPrime(ExactQuotient(Factorial(88),2^31) - 1);
true
> PrimalityCertificate(ExactQuotient(Factorial(88),2^31) - 1);
[* [* 86372085966821033095380685127128979368238229746650600193264699069266003642244357298640600053968141706854399999999999999999999, 1, 1, [* [* 83, 1 *], [* 79, 1 *], [* 73, 1 *], [* 71, 1 *], [* 67, 1 *], [* 61, 1 *], [* 59, 1 *], [* 53, 1 *], [* 47, 1 *], [* 43, 2 *], [* 41, 2 *], [* 37, 2 *], [* 31, 2 *], [* 29, 3 *], [* 23, 3 *], [* 19, 4 *], [* 17, 5 *], [* 13, 6 *], [* 11, 8 *], [* 7, 13 *], [* 5, 20 *], [* 3, 42 *], [* 2, 54 *] *], 17, [* 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5 *] *], [* 83, -1, 0, [* 83, 2, [* [* 2, 1 *], [* 41, 6, [* [* 2, 1 *], [* 5, 2, [* [* 2, 1 *] *] *] *] *] *] *] *] *]

According to the Magma handbook, IsPrime is a rigorous primality check (not a probabilistic one), and PrimalityCertificate gives a rigorous certificate for primality, coming from the ECPP (elliptic curve primality prover) algorithm.

Matthers
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