Constructions of space-frequency (SF) codes for multiple-input multiple-output (MIMO)-orthogonal frequency-division multiplexing (OFDM) systems with nt transmit antennas and Q subcarriers are considered in this paper. Following the pairwise-error-probability analysis, it is known that in addition to the conventional rank distance criterion, the minimum column distance of (nt × Q) SF codes serves as another benchmark in code design. SF codes with larger minimum column distance are expected to have better performance. Following this principle, the rate-diversity tradeoff for the MIMO-OFDM channels as well as two SF code constructions are presented. The first construction is obtained by right-multiplying the code matrices in a maximal rank-distance (MRD) code by a fixed (Q × Q) nonsingular matrix. Codes obtained from this construction are called linearly transformed MRD (LT-MRD) codes. Minimum column distance of the LT-MRD codes, when averaged over all code ensembles, is shown to meet the Gilbert-Varshamov bound. For the case of constructing the (2 × 256) quadrature phase-shift keying (QPSK)-modulated SF codes, it is shown that the LT-MRD codes can provide a much larger minimum column distance at the value of ≥ 50, compared to the values of 3; 5; or 6 obtained by other available constructions. The second code construction, termed cyclotomic construction, is reminiscent of the construction of the Reed-Solomon codes except that the code polynomials are now selected according to the cyclotomic cosets of the underlying field. Exact minimum rank distances of the resultant codes are presented. It is shown that this newly constructed code is asymptotically optimal in terms of rate-diversity tradeoff. Bounds on the minimum column distance of these codes are also given.
- Algebraic code designs
- Cyclotomic cosets
- Linearly transformed maximal rank distance (LT-MRD) codes
- Multiple-input multiple-output (MIMO)
- Orthogonal frequency-division multiplexing (OFDM)
- Space-frequency codes
- Space-time (ST) codes