Codes of \(C(n,n, 1)\) Designs

We associate codes with \(C(n,n,1)\) designs. The perfect \(C(n,n,1)\) designs obtained from perfect one-factorizations of \(K_n\) yield codes of dimension \(n-2\) over \(\mathbb{F}_2\) and \(n-1\) over \(\mathbb{F}_p\), for \(p\neq 2\). We also demonstrate a method of obtaining another \(C(n,n,1)\) design from a pair of isomorphic perfect \(C(n,n,1)\) designs and determine the dimensions of the resulting codes.