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On the wikipedia article about Hadamard Matrix it says that "The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is $92$"

But it also says that a new Hadamard matrix of size $nm$ can be created using Hadamard matrices of sizes $n$ and $m$.

Why isn't $23$ ($92=2 \times 2 \times 23$) the smallest size which cannot be created this way?

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    $\begingroup$ Isn't it because only sizes divisible by $4$ should be considered ? $\endgroup$
    – Denis Serre
    Commented Sep 25, 2020 at 14:17
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    $\begingroup$ That Wikipedia page also says "The order of a Hadamard matrix must be 1, 2, or a multiple of 4. " So the smallest size that can't be created is not $23$ but $3$. $\endgroup$
    – bof
    Commented Sep 25, 2020 at 14:22
  • $\begingroup$ The article states that using both Sylvester's and Paley's constructions, one cannot get Hadamard matrix of size 92 (As the first example of numbers divisible by 4 of course). Is that correct? why is that? If I can use primes like 2,3,5 I can also use 23. Can I construct those matrices without using any primes, and only starting with 1? @DenisSerre $\endgroup$
    – KfirKrak
    Commented Sep 25, 2020 at 16:47

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Using the Paley construction I, we obtain Hadamard matrices of size $4, 8, 12, 20, 24, 28, 32, 44, 48, 60, 68, 72, 80, 84, 88$. Using Paley Construction II we add $36=2(17+1)$, $52=2(25+1)$, $76=2(37+1)$. Using Sylvester's on the right sizes adds the sizes $16, 40, 56, 64$ which completes the list up to 88.

For 92, 91 is no prime power nor is $(92/2)-1=45$ so both Paley constructions give no result. And 46 is not a multiple of 4, so Sylvester also does not help. You cannot use Paley's on 23 without adding 1 dimension. For complex Hadamard matrices, the dimension can be a non-multiple of 4, it can be any integer.

If you're interested in (complex) Hadamard matrices or the Paley construction, take a look at my master thesis: Hadamard matrices over *-rings

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  • $\begingroup$ This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review $\endgroup$
    – Alex M.
    Commented Aug 24, 2022 at 9:45
  • $\begingroup$ It is not clear what this post attempts to answer, but definitely not the question. The OP asked why $23$ is not the smallest dimension of a Hadamard matrix with certain properties. The answer is given in the comments: Hadamard matrices have dimension $1$, $2$ or $4n$, with integer $n$, and $23$ is not such a number. $\endgroup$
    – Alex M.
    Commented Aug 24, 2022 at 9:50
  • $\begingroup$ OP changed his question to a new question in his last comment. That 23 is out of scope was already pointed out. $\endgroup$
    – Maarten Havinga
    Commented Aug 24, 2022 at 9:51
  • $\begingroup$ Ah, agreed. The OP's approach is a bit borderline though, since comments are not meant for new questions. $\endgroup$
    – Alex M.
    Commented Aug 24, 2022 at 9:53

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