Quasi-symmetric designs on $ 56 $ points

Abstract

<p style='text-indent:20px;'>Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with <inline-formula><tex-math id="M2">\begin{document}$ 56 $\end{document}</tex-math></inline-formula> points. One new <inline-formula><tex-math id="M3">\begin{document}$ (56,16,18) $\end{document}</tex-math></inline-formula> and many new <inline-formula><tex-math id="M4">\begin{document}$ (56,16,6) $\end{document}</tex-math></inline-formula> designs are discovered, and non-existence of <inline-formula><tex-math id="M5">\begin{document}$ (56,12,9) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ (56,20,19) $\end{document}</tex-math></inline-formula> designs with certain automorphism groups is proved. The number of known symmetric <inline-formula><tex-math id="M7">\begin{document}$ (78,22,6) $\end{document}</tex-math></inline-formula> designs is also significantly increased.</p>