Optimal rate-diversity tradeoff STBCs from codes over arbitrary finite fields

Abstract

Abstract-A linear rank-distance code is a set of matrices over a finite field Fq, with the rank over Fq as a distance metric. A Space-Time Block Code (STBC) is a finite set of complex matrices with the rank over the complex field as a metric. Rank-distance codes over prime fields Fp have found applications as space-time codes. In this paper, we extend this result to arbitrary finite fields by providing an isomorphism from Fq (q = p m ) to a subset of the ring of integers of an appropriate number field. Using this map and a maximal rank-distance code over Fq, we construct STBCs that achieve optimal rate-diversity tradeoff for any given diversity order. Simulation results confirm the diversity gain obtained using these codes.

Space–Time Codes From Sum-Rank Codes

“…Theorem 1 thus provides an equivalent definition of transmit diversity gain as a property of the code used. The recognition of the quantity (6) as a criterion for space-time code design first occurs in [24], [26] and the multiblock generalization which we adopt was first recognized in [7], [9], [25]. The equivalence to Definition 1 follows from [27], [28] which provide the required lower bound and state this equivalence.…”

“…From (6), we have that the transmit diversity gain d is an integer satisfying 1 ≤ d ≤ L • min{n t , T }. Codes for which d = L • min{n t , T } are referred to as being full diversity.…”

“…2) We provide sequential maximum likelihood (ML) decoders for these codes. More generally, we show that many sequential decoding strategies for space-time codes [10]- [16] that are typically thought to be only applicable to codes with a linear dispersion form [17] can in some cases be effectively adapted for use with the proposed codes and similarly constructed codes [3]- [6].…”

“…3) Facilitated by these ML decoders, we provide an empirical study of the performance of the proposed codes in simulation as well as related codes [3]- [6] which had not been previously decoded for large codebook sizes. This demonstrates that these codes can outperform full diversity codes [18]- [21] based on cyclic division algebras (CDAs) [22] at low SNRs and using smaller constellations.…”

Optimal STBCs from codes over galois rings

“…The size of the STBC so obtained is equal to the size of the code C GR . The Galois ring GR(p a , f ) is a finite field when a = 1, in which case, this construction reduces to the STBC construction from RD codes over arbitrary finite fields [14,15].…”