Log-Domain Calculation of the 1959 Sphere-Packing Bound with Application to M-ary PSK Block Coded Modulation

The 1959 sphere-packing bound of Shannon is reviewed, and a new algorithm is derived for its calculation. This algorithm performs the entire calculation in the logarithmic domain, thus facilitating the exact calculation of the bound for moderate to large block lengths. The new algorithm circumvents the numerical difficulties associated with a previously suggested algorithm by Valembois and Fossorier. We also compare the bound with a new sphere-packing bound which was recently derived by the authors. By applying the bound to M-ary PSK block coded modulation, the new bound suggests an interesting alternative to the Shannon bound.