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 \(\phi \).

具有交叉数\(x=1, y=3\)的准对称 2-(41, 9, 9) 设计是否存在是一个长期存在的悬而未决的问题。使用线性代码和子设计的性质，我们证明不存在循环准对称 2-(41, 9, 9) 设计，并且如果\(p<41\)是素数，则其自同构的阶数为准对称 2-(41, 9, 9) 设计，则\(p\le 5\). 与块交叉编号为 1 和 3 的准对称 2-(41, 9, 9) 设计的点相关的派生设计是块交叉编号为 0 的准对称 1-(40, 8, 9) 设计2.后者的关联矩阵生成长度为40的二进制双偶码。使用Betsumiya等人分类的长度为40的二进制双偶自对码数据库。(Electron J Combin 19(P18):12, 2012)，我们证明不存在具有恰好一个不动点的 5 阶自同构\(\phi \)的准对称 2-(41, 9, 9) 设计这样派生设计的二进制代码包含在\(\phi \)下的双偶自对偶 [40, 20] 代码不变式中。