Shortening Array Codes and the Perfect 1-Factorization Conjecture

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of Kp+1 and K 2 p , where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1 -factorization of K n is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K p+1 family of perfect 1 -factorization from the K 2p family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related.