Extending Divsalar´s bound to nonbinary codes with two dimensional constellations

Closed form upper bounds to error probabilities of nonbinary coded systems with two and higher dimensional constellation over the additive white Gaussian noise channel are derived. Computation of these bounds do not require integrations nor parameter optimizations. These bounds are the extension of bounds derived by Divsalar in 1999 for binary codes with binary modulations. The proposed bounds require only the knowledge of the average pairwise Euclidean distance enumerator of the code words when certain symmetry conditions do not hold. The bounds are tight for large block lengths. We also briefly discuss how to extend the pairwise Euclidean distance enumerators to ensembles of nonbinary protograph codes using the notion of frequency weight enumerators.