Shortening Array Codes and the Perfect -Factorization Conjecture

The existence of a perfect -factorization of the complete graph with nodes, namely, , for arbitrary even number , is a 40-year-old open problem in graph theory. So far, two infinite families of perfect -factorizations have been shown to exist, namely, the factorizations of and , where is an arbitrary prime number . It was shown in previous work that finding a perfect -factorization of 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 family of perfect -factorization from the family. Namely, techniques from coding theory are used to prove a new result in graph theory—that the two factorization families are related.