To determine the maximum rate that can be realized when a full spatial diversity is achieved, we need to define the optimal rate-diversity tradeoff [13] which can be characterized by the following equation:

where is the modulation symbol rate per channel use (p.c.u.), the quantity is termed the transmit diversity gain, and is the extension degree of the transmission symbol set drawn from the modulation symbol set. In our construction, a transmission symbol is drawn from a constellation set, thus . Moreover, a full transmit diversity is quantified by , therefore the optimal rate-diversity tradeoff implies a maximum rate of one modulation symbol p.c.u., that is, a full diversity STECC has a maximum rate equals to . As a consequence, we use for the construction -rate inner codes.

As the maximum rate of the inner code is determined, it is important to note that in the following for and , the selected inner codes have the maximum of the minimum Hamming distance for a combination of transmit antennas and time periods.

### 3.1. Full-Diversity STECC Designs

In order to realize the full spatial diversity, we recall that the binary rank criterion is a sufficient but not necessary condition to guarantee a full spatial diversity [12]. We apply the unified construction proposed in [13] to ensure a full spatial diversity for high-order modulations.

In [12], the authors prove that if every nonzero codeword of a linear binary code matrix has a maximum rank over the binary field , then for a binary phase shift keying (BPSK) transmission, the STECC achieves full spatial diversity, that is, . Moreover, it is demonstrated in [13] that if a STECC achieves full spatial diversity for BPSK transmission then using the Lu and Kumar construction (unified construction), we can obtain a full diversity STECC for high-order modulations based on full rank linear binary code matrices.

It yields a sufficient but not necessary condition on the linear binary inner code matrix to guarantee full diversity STECCs.

### 3.2. Threaded Layering Approach

To maximize the STECC diversity, we consider the threaded layering approach [5]. We assume rectangular STECCs of size . To design full diversity STB code, the threaded layering approach consists in splitting information symbols into disjoint threads. The threads must be active over the transmission intervals. For each layer (thread) and at each transmission interval, symbols of this layer are transmitted. Threads use equally often the transmit antennas. To ensure a maximum diversity, each thread must achieve a maximum diversity when symbols corresponding to the other threads are put to zero and threads must be transparent to each other. This can be realized by affecting weighting numbers to each thread such that resulting threads span disjoint algebraic subspaces. These numbers are "Diophantine number".

In the case of STECCs, linear combinations are applied on binary elements which greatly relaxes the constraints to achieve full diversity STECCs. The number of threads is taken equal to the number of transmit antennas. The threaded layering set is defined, for , by

where denotes the mod- operation. Table 1 shows the threaded layering for and structures using 2 threads. We associate the information symbols (resp., redundancy symbols) to the first layer (resp., second layer). Let us denote by the information symbol vector and by the vector associated to the redundancy symbols. is an integer permutation matrix such that the STECC, built up from , , and a diophantine number, achieves a full spatial diversity ( must also keep properties of STECCs defined in (6), i.e., each entry of a space-time codeword matrix is composed of one modulation symbol).

Table 1 The threaded layering in coherent scenario using 2 layers. The numbers refer to thread indexes. The vertical and horizontal axes correspond to the spatial and temporal dimensions, respectively.

#### 3.2.1. Full-Diversity STECC

In this case, it was verified in [19] that a full diversity STECC can be defined by

where , whatever and (to ensure an energy efficiency). As the determinant of the difference codeword matrix depends on the value of , thus to maximize the coding gain we must carefully select the diophantine number. In [19], it was proved that is the optimal value for 4-QAM and 16-QAM. In this paper, by applying the binary rank criterion and the unified construction of [13] we verify that a full spatial diversity can be achieved without the necessity of a diophantine number (). For , (the same result can be obtained for the other possible 3-tuple) the associated linear binary code matrix is defined by

It clearly appears that the first and the second row of are linearly independent over the binary field . Thus for a BPSK transmission, the STECC constructed from achieves a full spatial diversity. Therefore, using the unified construction presented in [13], the STECC built up from this linear binary code matrix ensures a full spatial diversity for any order QAM modulation. We also note that this construction can be extended to phase shift keying (PSK) and pulse amplitude (PAM) modulations.

#### 3.2.2. Full-Diversity STECC

The inner code corresponds to the extended Hamming code. Due to the fact that the all-one vector is a codeword, the binary rank criterion is false whatever the spatial arrangement into a binary matrix. We thus apply the threaded layering approach.

The binary matrix is

and the associated space-time error correcting codeword is

To ensure full diversity diophantine number is chosen such that and . In that case, the maximization of the coding gain yields an optimum value for depending of the modulation order. However, in practice the STECC will be serially concatenated to an outer FEC and it was shown in [20] that the asymptotic global coding gain is independent of the choice of the parameter provided that and . In [21], we proposed a way to construct full diversity STECCs defined from an inner half-rate invertible linear binary codes.