Low density MDS codes and factors of complete graphs

We reveal an equivalence relation between the construction of a new class of low density MDS array codes, that we call the B-code, and a combinatorial problem known as perfect one-factorization of complete graphs. We use known perfect one-factors of complete graphs to create constructions and decoding algorithms for both the B-code and its dual code. The B-code and its dual are optimal in the sense that (i) they are MDS, (ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by the change of a single information bit is minimal and (iii) they have optimal length. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 year long conjecture in graph theory. The construction of B-codes of arbitrary odd length provides an affirmative answer to the conjecture.