Abstract
The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers \(x=1, y=3\) is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if \(p<41\) is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then \(p\le 5\). The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism \(\phi \) of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under \(\phi \).
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Munemasa, A., Tonchev, V.D. Quasi-symmetric 2-(41, 9, 9) designs and doubly even self-dual codes of length 40. AAECC 33, 855–866 (2022). https://doi.org/10.1007/s00200-022-00543-w
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DOI: https://doi.org/10.1007/s00200-022-00543-w