An n xl (l_> n) space time block code (STBC) C over a complex signal set S con- sists of a finite number of n x I matrices with elements from $. For quasi-static, fiat fading channels a primary performance index of C is the minimum of the rank of the difference of any two codeword matrices, called the rank of the code. C is of full-rank if its rank is n and is of minimal-delay if l = n. If the rank of an n x l STBC over S is r then the rate of transmission R for this code in bits per second per Hertz is up- 1 per bounded by T 10g2 where denotes the maximum size of a code of length y and minimum Hamming distance z defined over an alphabet of size x. From this it follows that for full-rank codes over S, R is upper bounded by 10g2 ISI . A code over S meeting the upper bound on R is called a full-rate code. A minimal-delay full-rank, full-rate STBC over S is called optimal over $. Working with cycloromic field extensions of the rational field several families of codes over a wide range of signal sets can be constructed [18, 20]. As spe- cial cases, our method gives optimal STBCs over symmetric m-PSK signal sets (m-arbitrary) for a wide range of values of the number of antennas n. In this paper, we show that all such STBCs admit ML decoding with complexity linear in the size of the signal set m, when the number of received is 1, whereas for in general the decoding complexity is exponential in m.

An n l (l n) space time block code (STBC) C consists of a nite number jCj of n l matrices with entries from the complex eld. If the entries of the codeword matrices take values from a complex signal set S or complex linear combinations of elements of S then the code is said to be over S. For quasi-static, at fading channels a primary performance index of C is the minimum of the rank of the dierence of any two codeword matrices, called the rank of the code. C is of full-rank if its rank is n and is of minimal-delay if l = n. The rate B.A.Sethuraman is with AUKBC Research Center, Anna University|M.I.T. Campus, Chennai 600044, India, and Dept. of Mathematics, California State University Northridge, CA 91330, U.S.A., email:al.sethuraman@csun.edu B.S.Rajan is with the Department of Electrical Communication Engineering and the Department of Computer Science and Automation, Indian Institute of Science, Bangalore560 012, India, email:bsrajan@ece.iisc.ernet.in This was partly supported by a generous grant from K.B.Chandrashekhar Family Foundation to the rst author and by Academic and Research Collaboration Programme on Mathematical Engineering between the DRDO, New Delhi and IISc, Bangalore, India, to the second author.