2 where s is the receiver input power and n b is the received gaussian noise in the bit rate bandwidth for 64 qam with a gray coded modified square array constellation as indicated in either fig3 or 4 the probability of bit error p be 64 qam ms is computed by the inventor to be p be u2061 64 qam u2062 ms 13 24 u2062 q u2061 6 23 u2062 s n b 1 2 what the above results indicate is that with gary coding for a given bit rate a given bit error rate and when the constellation is a modified square as embodied in this invention the required received signal level is approximately 0 35 db more than that required when the constellation is square given this degradation in receiver performance use of the modified square constellation in a 64 qam system results in a net system gain improvement of 0 84 db 1 19 db 0 35 db over a 64 qam system using a square gray coded constellation and possibly a slightly different improvement if the square constellation is not gray coded for 256 qam with a gray coded square array constellation the probability of bit error p be 256 qam s is shown in the text u201c fixed broadband wireless communications u201d prentice hall 2004 by douglas h morais to be given by p be u2061 256 qam u2062 s 15 32 u2062 q u2061 8 85 u2062 s n b 1 2 for 256 qam with a gray coded modified square array constellation as indicated.