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I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?

According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard matrices and Seberry and Yamada - Amicable Hadamard matrices and amicable orthogonal designs) this matrix has been constructed. However, none of the papers I have found gives a reference to the actual construction that is being used to obtain it.

From what I have seen, it looks like the first paper which mentions its existence was published in 1978 (Seberry - On skew Hadamard matrices), with the first (non-skew) construction of an Hadamard matrix of order 292 being published in 1975 (Spence - Skew-Hadamard Matrices of the Goethals–Seidel Type).

The idea is to make constructions for various types of Hadamard matrices available in SageMath. We currently have quite a few already (and discovered gaps in literature along the way).

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  • $\begingroup$ @AlexeyUstinov I've edited the question to make it clearer. $\endgroup$
    – Matteo Cati
    Commented Jan 23, 2023 at 17:42
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    $\begingroup$ @LSpice Full disclosure: the poster is a student and a collaborator of mine. $\endgroup$
    – Dima Pasechnik
    Commented Jan 23, 2023 at 22:46
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    $\begingroup$ From the proof of Prop. 3.4 in arxiv.org/abs/1008.2043: "there is an infinite series of skew Hadamard matrices constructed by E. Spence [16] which contains such a matrix of order 4·73." The reference points to the paper cited in OP's question. $\endgroup$
    – Steve Huntsman
    Commented Jan 24, 2023 at 13:44
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    $\begingroup$ @SteveHuntsman in [16] there is a construction of a Hadamard matrix of order 292, but it's not skew-symmetric (and it is not claimed there in [16] to be so, either). $\endgroup$
    – Dima Pasechnik
    Commented Jan 24, 2023 at 15:40
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    $\begingroup$ @SteveHuntsman - Djokovic acknowledged that his reference to [16] was not correct - but sent us his own construction that fills the gap (for 292). $\endgroup$
    – Dima Pasechnik
    Commented Jan 26, 2023 at 16:45

1 Answer 1

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The skew Hadamard matrix of order 292 can be constructed from skew supplementary difference sets (kindly supplied to us by Prof. Djokovic). This same construction is used, for example, in Djokovic - Skew-Hadamard Matrices of Orders 436,580, and 988 Exist.

An implementation of this can now be found in SageMath.

Note that $H=\{1,2,4,8,16,32,64,55,37\}$ is the order 9 subgroup of $Z_{73}$, and let $$ J_1 = \{5,9,11,25\}\\ J_2 = \{11,13,17,25\}\\ J_3 = \{5,9,13,17\}\\ J_4 = \{1,3,13\}\\ $$

Then, we can obtain supplementary difference sets $(X_1,X_2,X_3,X_4)$ with parameters $(73; 36,36,36,28; 63)$ and with block $X_1$ skew as follows: $$ X_1 = \bigcup_{x\in J_1} xH\\ X_2 = \bigcup_{x\in J_2} xH\\ X_3 = \bigcup_{x\in J_3} xH\\ X_4 = \{0\} \cup \bigcup_{x\in J_4} xH\\ $$

From each set $X_i$, let $a_i = (a_{i,0}, a_{i,1},...,a_{i,72})$ be the {±1}-row vector such that $a_{i,j} = −1$ iff $j \in X_i$.

Then, we can construct four circulant matrices $A_1, A_2, A_3, A_4$ where $a_i$ is the first row of $A_i$. We have that $A_1$ is skew, and by plugging them in the Goethals-Seidel array we obtain a skew Hadamard matrix of order 292: $$ H = \left(\begin{array}{rrrr} A_1 & A_2R & A_3R & A_4R \\ -A_2R & A_1 & -A_4^TR & A_3^TR \\ -A_3R & A_4^TR & A_1 & -A_2^TR \\ -A_4R & -A_3^TR & A_2^TR & A_1 \end{array}\right) $$

Here $R$ denotes the matrix having ones on the back-diagonal and all other entries zero.

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