An edge coloring of a graph G is a mapping Á : EG ! N. The edge coloring Á is called strong if Áe 6= Áe0 for any two edges e and e0 that are distance at most one apart. The minimum number of colors needed for a strong edge coloring of a graph G is called strong chromatic index of G and denoted by Â0 sG. Strong edge colorings of graphs were introduced by Fouquet and Jolivet [1]. Clearly, Â0 sG cedil; cent;G for any graph G. In 1985, during a seminar in Prague, Erd˝os and Ne˘set˘ril put forward the following conjecture.Conjecture 1. For every graph G with maximum degree cent;,formulaErd˝os and Ne˘set˘ril provided a construction showing that Conjecture 1 is tight if itrsquo;s true. In 1997, using probabilistic method, Molloy and Reed [3] showed that Â0 sG middot; 1:9982cent; for a graph G with sufficiently large cent;. The currently best known upper bound for a graph G is 1:932cent;, due to Bruhn and Joos [2]. The Hamming graph Hn;m is the Cartesian product of n copies of the complete graph Km n;m 2 N. Theorem 1. Let Hn;m be a Hamming graph with m cedil; 2. Thenformulaand the upper bound is sharp for m = 2. Theorem 2. Let Hn; 3 be a Hamming graph. Thenformulaand the upper bound is sharp for n = 3.