Computing the performance of unitary space-time group codes from their character table

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment. Their success, however, depends on the design of cedes that achieve these promises. It is well known that unitary matrices can be used to design differentially modulated space- time codes. These codes have a particularly efficient description if they form a finite group under matrix multiplication. We show how to compute the parameters of such groups crucial for their use as space-time codes, using only the character table of the group. Since character tables for many groups are known and tabulated, this method could be used to quickly test, for a given group, which of its irreducible representations can be used to design good unitary space-time codes. We demonstrate our method by computing the eigenvalues of all the irreducible representations of the special linear group $SL_2(F_q)$ over a finite prime field $F_q$ of odd characteristic, and study in detail the performance of a particular eight-dimensional representation of $SL_2(F_17)$