Quasi-symmetric \(2\)-\((28,12,11)\) designs with an automorphism of order \(5\)

A design is called quasi-symmetric if it has only two block intersection numbers. Using a method based on orbit matrices, we classify quasi-symmetric \(2\)-\((28,12,11)\) designs with intersection numbers \(4\), \(6\), and an automorphism of order \(5\). There are exactly \(31\,696\) such designs up to isomorphism.