Naslov
Two Series of Regular Hadamard Matrices

Autori
Crnković, Dean

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Combinatorics 2006 / - Lahti, 2006

Skup
Combinatorics 2006

Mjesto i datum
Ischia, Italija, 25.06.2006. - 02.07.2006

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
regular Hadamard matrix; symmetric design; Menon design

Sažetak
T. Xia, M. Xia and J. Seberry proved (see [2]) the following statement: When $k=q_1, q_2, q_1q_2, q_1q_4, q_2q_3N, q_3q_4N$, where $q_1, q_2$ and $q_3$ are prime powers, $q_1 \equiv 1\ (mod \ 4)$, $q_2 \equiv 3\ (mod \ 8)$, $q_3 \equiv 5\ (mod \ 8)$, $q_4=7$ or $23$, $N=2^a3^bt^2$, $a, b=0$ or $1$, $t \neq 0$ is an arbitrary integer, there exist regular Hadamard matrices of order $4k^2$. The existence of some regular Hadamard matrice of order $4p^2$, when $p$ is a prime, $p \equiv 7\ (mod \ 16)$, is established in [1]. According to [1] and [2], there are just two values of $k \le 100$ for which the existence of a regular Hadamard matrix of order $4k^2$ is still in doubt, namely $k=47$ and $k=79$. We prove the following assertion: Let $p$ and $2p-1$ be prime powers and $p \equiv 3\ (mod\ 4)$. Then there exists a symmetric design with parameters $(4p^2, 2p^2 - p, p^2 - p)$. Thus there exists a regular Hadamard matrix of order $4p^2$. In particular, there exists a regular Hadamard matrix of order $4 \cdot 79^2=24964$. If $p$ and $2p-1$ are primes, then a derived designs of the symmetric $(4p^2, 2p^2 - p, p^2 - p)$ design is cyclic. In a similar way we construct a symmetric $(4(p+1)^2, 2p^2 +3p+1, p^2 + p)$ design ${; ; \cal D}; ; $ for prime powers $p$ and $2p+3$, such that $p \equiv 3\ (mod\ 4)$. If $p$ and $2p+3$ are primes, then a derived design of ${; ; \cal D}; ; $ is 1-rotational. REFERENCES: [1] K.~H. Leung, S.~L. Ma and B.~Schmidt, New Hadamard matrices of order $4p^2$ obtained from Jacobi sums of order 16, preprint. [2] T.~Xia, M.~Xia, and J.~Seberry, Regular Hadamard matrices, maximum excess and SBIBD, Australasian Journal of Combinatorics, Vol. 27 (2003) pp. 263--275.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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