Unitary Space–Time Group Codes: Diversity Sums From Character Tables

Diversity sum, which is calculated from the Frobenius norm of the difference of two distinct elements in a signal constellation, is the significant parameter to predict a unitary space–time constellation having good performance in low signal-to-noise ratio (SNR). In this correspondence, we propose a method to compute the diversity sum of a unitary group constellation using a character table. Our proposed analysis is simple, requiring only a lookup of the character table. We illustrate our method for the finite special linear groups $SL_{2}$, and compare codes with high diversity sum against fixed point free groups at low SNR. We also introduce the notion of a faithful group constellation, that is, one whose diversity sum is greater than $0$. Faithful group constellations may be obtained from any nontrivial character of a group. We describe the method to do so in this correspondence and illustrate it with the example of the finite projective special linear group ${\rm PSL}_{2}$.