Horton, J.D., A hyperfactorization of order 8, index 2, Discrete Mathematics 127-129. 92 (1991) Consider the complete graph on 2n vertices, Kzn. A one-factor of Kti is a set of n edges (pairs of vertices) such that each vertex is in exactly one of the edges. A hyperfactorization of index il, or A-hyperfuctorization, is a family of one-factors such that each pair of edges without vertices in common occur together in exactly rl of the one-factors. A hyperfactorization is said to be simpfe if all the one-factors are distinct. A trivial hyperfactorizzt!n;i can be constructed by taking the family of a:i one-factors. Each pair of edges occur together in (2n 5)(2n 7) l l l (1) one-factors, the number of one-factors in Kb_+ In particular for Kg, n = 4, the trivial hyperfactorization has index 3. The only known nontrivial hyperfactorizations of index 1 are due to Cameron [2], where the orders are of the form 2n = 2“ + 2, CI 2 3. Boros, Jungnickel and Vanstone have [l] constructed recursively non-trivial simple A-hyperfactorizations of Kti for all II 2 5, but with much larger indexes. Jungnickel and Vanstone [3] have constructed two other examples, 2n = 12, A = 15, and 2n = 24, A = 495. These are the only known constructions. Two non-existence results are that no hyperfactorization of index 1 exists for KS (Cameron [2], Mathon [5]), nor for Klz (Lam et al., [4]). In this papar, we ar, p interested only in K8 and k = 2. Clearly, a simple 2-hyperfactorization does not exist, for if one did, the complementary hyperfactorization with respect to the trivial hyperfactorization (A = 3) would be a hyperfactorization of index 1. However, a nonsimple hyperfactorization of KH