Construction of New Delay-TolerantSpace-Time Codes

# Construction of New Delay-Tolerant Space-Time Codes

Mireille Sarkiss, Ghaya Rekaya-Ben Othman, Mohamed Oussama Damen
and Jean-Claude Belfiore
###### Abstract

Perfect Space-Time Codes (STC) are optimal codes in their original construction for Multiple Input Multiple Output (MIMO) systems. Based on Cyclic Division Algebras (CDA), they are full-rate, full-diversity codes, have Non-Vanishing Determinants (NVD) and hence achieve Diversity-Multiplexing Tradeoff (DMT). In addition, these codes have led to optimal distributed space-time codes when applied in cooperative networks under the assumption of perfect synchronization between relays. However, they loose their diversity when delays are introduced and thus are not delay-tolerant. In this paper, using the cyclic division algebras of perfect codes, we construct new codes that maintain the same properties as perfect codes in the synchronous case. Moreover, these codes preserve their full-diversity in asynchronous transmission.

Cooperative Communication, Distributed Space-Time Codes, Perfect Codes, Delay-Tolerance, Cyclic Division Algebra, Tensor product.

## I Introduction and Problem Statement

During the past decade, MIMO techniques have experienced a great interest in wireless communication systems. Using multiple antennas at the transmitter and the receiver provides high data rates and exploits the spatial diversity in order to fight channel fadings and hence improve the link reliability. Lately, cooperative diversity has emerged as a new form of spatial diversity via cooperation of multiple users in the wireless system [1]. While preserving the same MIMO benefits, it counteracts the need of incorporating many antennas into a single terminal, especially in cellular systems and ad-hoc sensor networks, where it can be impractical for a mobile unit to carry multiple antennas due to its size, power and cost limitations.

In cooperative networks, users communicate cooperatively to transmit their information by using distributed antennas belonging to other independent terminals. This way, a virtual MIMO scheme is created, where a transmitter is also acting as a relay terminal, with or without some processing, assisting another transmitter to convey its messages to a destination. The cooperative schemes have been widely investigated by analyzing their performance through different cooperative protocols [1, 2, 3]. These protocols fall essentially into two families: Amplify-and-Forward (AF) and Decode-and-Forward (DF). In order to achieve the cooperative diversity, space-time coding techniques of MIMO systems have also been applied yielding many designs of distributed space-time codes under the assumption of synchronized relay terminals [2, 3, 4].

However, this a priori condition on synchronization can be quite costly in terms of signaling and even hard to handle in relay networks [5, 6]. Unlike conventional MIMO transmitter, equipped with one antenna array using one local oscillator, distributed antennas are dispersed on different terminals, each one with its local oscillator. Thus, they are not sharing the same timing reference, resulting in an asynchronous cooperative transmission.

On the other hand, in a synchronous transmission, the distributed STCs are constructed basically according to the rank and determinant criteria [7] and hence aim at achieving full diversity. Note that the rows of the codeword matrix represent the different relay terminals (antennas). So, when asynchronicity is evoked, delays are introduced between transmitted symbols from different distributed antennas shifting the matrix rows. This matrix misalignment can cause rank deficiency of the space-time code, and thus performance degradation.

Therefore, the codes previously designed are no more effective unless they tolerate asynchronicity. Furthermore, an efficient code design should satisfy the full-diversity order for any delay profile. This intends to guarantee full-rank codewords distance matrix i.e., its rank equal to the number of involved relays, hence leading to the so-called delay-tolerant distributed space-time codes [6].

## Ii Delay-Tolerant Distributed Space-Time Codes

The first designs of such codes were presented by Li and Xia [6] as full-diversity binary Space-Time Trellis Codes (STTC) based on the Hammons-El Gamal stacking construction, its generalization to Lu-Kumar multilevel space-time codes, and the extension of the latter codes for more diverse AM-PSK constellations [8, 9]. Systematic construction including the shortest STTC with minimum constraint length was also proposed in [10], as well as some delay-tolerant short binary Space-Time Block Codes (STBC) [11]. Recently, Damen and Hammons extended the Threaded Algebraic Space-Time (TAST) codes to asynchronous transmission [12]. The delay-tolerant TAST codes are based on three different thread structures where the threads are separated by using different algebraic or transcendental numbers that guarantee a non-zero determinant of the codewords distance matrix. An extension of this TAST framework to minimum delay length codes was considered in [13].

Meanwhile, perfect space-time block codes that are optimal codes originally constructed for MIMO systems [14, 15, 16, 17], were also investigated for wireless relay networks. In [18, 19], the authors provided optimal coding schemes in the sense of DMT tradeoff [20], based cyclic division algebras for any number of users and for different cooperative strategies. Nevertheless, all these schemes assumed perfect synchronization between users. Then, it was in [21] that Petros and Kumar discussed the delay-tolerant version of the optimal perfect code variants for asynchronous transmission. They stated that delay-tolerant diagonally-restricted CDA codes and delay-tolerant full-rate CDA codes can be obtained from previous designs by multiplying the codeword matrix by a random unitary matrix. This matrix can be taken specifically from an infinite set of unitary matrices that do not have elements in the code field.

In this paper, we construct delay-tolerant distributed codes based on the perfect codes algebras from a different point of view. The new construction is obtained from the tensor product of two number fields, one of them being the field used for the perfect code. The codes are designed in such a way to maintain the same properties of their corresponding perfect codes in the synchronous transmission, namely full-rate, full-diversity and non-vanishing minimum determinant. In addition, unlike the perfect codes, the new codes preserve the full diversity in the asynchronous transmission.

## Iii Background

Before addressing the STC construction, we dedicate this section to briefly review the remarkable properties of the perfect codes as analyzed in [14, 15, 16, 17]. Then, following the framework of [6], we present the cooperative communication model of interest.

### Iii-a Perfect Space-Time Block Codes

The concept of Perfect Code was originally proposed in [14, 15] for transmit antennas to describe a square linear dispersion STC . The perfect codes are constructed from cyclic division algebras of degree defined by

1. and are number fields and the corresponding ring of integers. is called the base field and taken as or since the ST code transmits -QAM or -HEX information symbols for or , respectively. Thus, the constellations can be seen as finite subsets of the ring of Gaussian integers or Eisenstein integers , respectively.

2. is a cyclic Galois extension of of degree with or a field extension appropriately chosen in order to get an existing lattice and a division algebra, and an algebraic number.

3. is the generator of the Galois group , . For an element , the conjugates of are . So, the norm and the trace are defined respectively as

 NK/F(x)=n∏k=1σk(x),   TrK/F(x)=n∑k=1σk(x). (1)
4. the set of non-zero elements of . It is a non-norm element suitable for the cyclic extension [15].

The cyclic division algebra is then expressed as a right -space

 A1=K⊕eK⊕e2K⊕…⊕en−1K (2) with  e∈A,en=γ∈K,γ≠0  and  λe=eσ(λ)  for all  λ∈K. (3)

The Perfect Codes satisfy the criteria:

1. Full-rate: The code transmits symbols drawn from QAM or HEX constellation and thus has a rate of symbols per channel use (spcu).

2. Full-diversity: According to the rank criterion [7], the determinant of the codeword distance matrix for any two distinct codewords is non-zero. By code linearity, it can be reduced to

 (4)
3. Non-vanishing minimum determinant: The minimum determinant of any codeword distance matrix, prior to SNR normalization, is lower bounded by a constant that is independent of the constellation size

 δ(C)=min0≠X∈C|det(X)|2≥ψ>0 (5)
4. Cubic shaping: The QAM or HEX constellations are normalized according to the power at the transmitter so that the real vectorized codeword vectors are isomorphic to cubic lattices or . In other words, the rotation matrix encoding the information symbols into each layer is required to be unitary to guarantee the energy efficiency of the codes. The shaping constraint leads thus to two other properties. The first one is the Uniform average transmitted energy per antenna. The second one is the Information losslessness as the unitary linear dispersion matrix allows to preserve the mutual information of the MIMO channel.

Thanks to prominent results on diversity-multiplexing tradeoff [20], the perfect codes also verify two other equivalent properties:

1. DMT optimality: In [16], Elia et al.  proved that the full-rate STCs from cyclic division algebra having NVD property achieve the optimal DMT in Rayleigh fading channel.

2. Approximate universality: Being CDA-based codes with NVD property, the perfect codes are approximately universal and achieve DMT for arbitrary channel fading distribution.

Satisfying all these criteria, the perfect codes showed to improve the performance in terms of error probability upon the best known codes.

### Iii-B Cooperative System Model

In the sequel, we consider a cooperative system with a source communicating to a destination via relays in two phases as in Figure 1, and without direct links between the source and the destination. In the first phase, the source broadcasts its message to the potential relays. In the second phase, the relays use the DF protocol to detect the source message then if successfully detected transmit it to the destination. We assume that all the relays are able to achieve error free decoding which could be possible by selecting the source-relays links, and consider only the links that are not in outage. Note that it could also be possible that not all the relays may successfully decode the original message, so the number of transmitting relays is usually assumed as a random variable. Since the relays transmission overlap in time and frequency, they can cooperatively implement a distributed space-time code.

Considering only the second phase of transmission, the system is equivalent to a MIMO scheme where the distributed perfect space-time code is used by the relays, with transmit antennas one by relay, and receive antennas at the destination. Every time slot , the relays send the column vector of the codeword and the destination receives

 Yt=HtXt+Wt,   Yt,Wt∈CNr×1 (6)

where is the additive white Gaussian noise with i.i.d complex Gaussian variables with zero-mean and variance , , , being the noise variance per real dimension. represents the complex channel matrix modeled as i.i.d Gaussian random variables with zero mean and unit variance . The channel is assumed quasi-static with constant fadings during a transmitted codeword and independent fadings between subsequent codewords. Dealing with square STCs , the codeword matrix contains information symbols carved from two-dimensional QAM or HEX finite constellations denoted by .

### Iii-C Asynchronous Cooperative Diversity

The above expression (6) is valid only when relays are synchronized. In the presence of asynchronicity, the codeword transmission is spanned on more than symbol intervals due to delays. Although the symbol synchronization is not required, we assume that the relays are synchronized at the frame-codeword level, which can be provided by means of network feedback signaling from the destination. Therefore, the start and the end of each codeword are aligned for different relays by transmitting zero symbols, and hence there is no interference between codewords transmission. We further assume that the timing errors between different relays are integer multiples of the symbol duration and the fractional timing errors are absorbed in the channel dispersion. In the codeword matrix, these delays are also filled with zeros; they are known at the receiver but not at the transmitting relays [6].

Denoting a delay profile by , a delay corresponds to the relative delay of the received signal from the relay as referenced to the earliest received relay signal. Let denotes the maximum of the relative delays, then from the receiver perspective, the codeword matrix was sent instead of the space-time code.

### Iii-D Motivation of the Code Construction

The diversity order of any space-time code is defined by the minimum rank of the distance codeword matrix over all pairs of distinct codewords [7]. The distributed perfect codes are full-rate full-diversity for the synchronous transmission between the relays and the destination. Note that in general, a transmission between source, half-duplex relays and destination will result in rate loss. When asynchronicity is introduced, the code is no more full-rate since it is spanned on time instants. Moreover, certain delay profiles can result in linearly dependent rows, thus the code will loose its full-diversity property. Let us illustrate this by the following example.

##### Example of Golden Code

We consider the distributed Golden code transmitting information QAM symbols from two synchronized relays with the codeword matrix.

 Xs=1√5[α(s1+s2θ)α(s3+s4θ)i¯α(s3+s4¯θ)¯α(s1+s2¯θ)] (7)

The Golden code is designed on a cyclic field extension of degree over the base field . Using the generator matrix of the corresponding complex -dimensional lattice, the codeword elements are lattice points obtained by linear combination of pairs of symbols.

Now, let the first relay be delayed by one symbol period with respect to the second , such that the new asynchronous codeword matrix be

 Xa=1√5[0α(s1+s2θ)α(s3+s4θ)i¯α(s3+s4¯θ)¯α(s1+s2¯θ)0] (8)

Suppose we have two distinct codewords and with and the other symbols equal i.e., . The difference between matrix codewords is defined in both synchronous and asynchronous cases as

 Δ(s)s=[2αs1002¯αs1] ,   Δ(s)a=[02αs1002¯αs10] (9)

It can be seen that is a full-rank matrix whereas has rank one, so the Golden code is not a delay-tolerant code.

In fact, it can be seen from the asynchronous codeword matrix that some symbols are aligned at the same instant due to delays loosing thus diversity. In order to resolve this problem of rank deficiency, our solution consists in transmitting from each antenna (relay) at each transmission time a different combination of all the information symbols. This way, in the presence of delays, we ensure that any combined symbol sent from the relays arrives at the destination in at least different instants, hence guaranteeing the full-diversity order of the space-time code.

A new STC will have then the shifted codeword matrix

 Xa=[0f1(s1,s2,s3,s4)f2(s1,s2,s3,s4)f3(s1,s2,s3,s4)f4(s1,s2,s3,s4)0]. (10)

Now, to get these linear combinations of the symbols, we need a higher dimensional lattice compared to the -dimensional lattice used for the Golden code. So, we propose to obtain the corresponding lattice generator matrix by the tensor product of two field extensions of , one of them being the field extension of the Golden code.

Following this idea, we aim at constructing, in general, new codes that are based on CDA of the perfect codes such that they maintain the same optimal properties as perfect codes in the synchronous case. But also, these codes preserve their full-diversity in asynchronous transmission and thus are delay-tolerant for arbitrary delay profile.

## Iv Construction of Delay-Tolerant Distributed Codes based Perfect Codes Algebras

### Iv-a General Construction

The approach consists in constructing a division algebra isomorphic to the tensor product (also called Kronecker product or cross-product) of two number fields of lower degrees. Other constructions based on the crossed-product algebras have been investigated in [22, 23] either for prime or coprime degrees of the composite algebras. In these constructions, the space-time code was built on the cyclic product algebra. However, in the present construction, the higher degree algebra is only used to derive appropriately the space-time code.

Since we intend to construct a full-rate space-time code that is based on the CDA of the full-rate perfect code, then the first algebra to be considered is the cyclic division algebra of the perfect code of degree over the base field . For sake of simplicity, we analyze in the sequel the case of Gaussian Field to explain the construction. Indeed, we consider the cyclic field extension of degree over , being an algebraic number. The principal ideal is generated by an element and its integral basis is (or if unitary, it is given by ). The basis of the complex algebraic lattice is obtained by applying the canonical embedding to . Consequently, the generator matrix corresponds to the rotation matrix in

 M1=1√p1⎡⎢ ⎢ ⎢ ⎢ ⎢⎣v1v2…vMσ1(v1)σ1(v2)…σ1(vM)⋮⋮⋱⋮σM−11(v1)σM−11(v2)…σM−11(vM)⎤⎥ ⎥ ⎥ ⎥ ⎥⎦, (11)

where is a normalization factor used to guarantee the matrix unitarity.

Now, we consider another Galois extension over of the same degree such that its discriminant is coprime to the one of i.e., . Let with an algebraic number. The Galois group is generated by as . The principal ideal of the algebra is such that and thus its integral basis is given by . The canonical embedding of gives another complex rotated lattice of that is generated by the unitary matrix with the normalization factor,

 M2=1√p2⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1θ2…θM−121σ2(θ2)…σ2(θM−12)⋮⋮⋱⋮1σM−12(θ2)…σM−12(θM−12)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (12)

The tensor product of both field extensions allows to build a rotated lattice in higher dimension corresponding to the complex unitary matrix based on the previous constructions. According to [24],

##### Proposition 1

: Let be the compositum of the above Galois extensions, of order over as presented in Figure 2.

Since and have coprime discriminants, the corresponding lattice generator matrix can be obtained as the tensor product of the previous unitary generator matrices.

 M=M2⊗M1=1√p1p2⋅
 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣v1⋯vM⋯v1θM−12⋯vMθM−12⋮⋱⋮⋱⋮⋱⋮σM−11(v1)⋯σM−11(vM)⋯σM−11(v1)θM−12⋯σM−11(vM)θM−12⋮⋱⋮⋱⋮⋱⋮v1⋯vM⋯v1σM−12(θM−12)⋯vMσM−12(θM−12)⋮⋱⋮⋱⋮⋱⋮σM−11(v1)⋯σM−12(vM)⋯σM−11(v1)σM−12(θM−12)⋯σM−11(vM)σM−12(θM−12)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦\vspace−10pt (13)

Consequently,

##### Proposition 2

: Let the order of the extensions, then the discriminant of is . The minimum product distance of the lattice is derived from the discriminant of as

 dp,min=1√dK=1√dm1K1dm2K2 (14)

Using the matrix , the space-time coded components are given by the linear combination where is the information symbol vector carved from a -QAM constellation . Then, the space-time codeword matrix is defined by distributing the components with appropriate constant factors . It can be represented as a Hadamard product

 X = [Φ]∙[x]=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ϕ1x1ϕM+1xM+1⋯ϕM(M−1)+1xM(M−1)+1ϕ2x2ϕM+2xM+2⋯ϕM(M−1)+2xM(M−1)+2⋮⋮⋱⋮ϕMxMϕ2Mx2M⋯ϕM2xM2⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ (15)

The key idea in the code construction is to determine the coefficients that allow one to preserve the same properties of the corresponding perfect codes in synchronous transmission (Section III-A).

1. On one side, it can be seen that the new code transmits information symbols and thus is full-rate with spcu for a relays-destination transmission phase.

2. On the other side, we need to find the factors that satisfy the rank criterion (4) in order to have full-diversity codes.

3. Moreover, the perfect codes have non-vanishing minimum determinants. Then, we are interested in deriving ST codes that have not only non-zero determinants, but also these determinants do not vanish when constellation size increases.

4. In order to guarantee uniform energy distribution in the codeword, we ask that verify . Choosing further the coefficients yields better determinants as obtained for the non-norm elements of the perfect codes [15]. This restricts the values of to .

5. It can also be noticed that the new code satisfies the cubic shaping property since the generator matrix of the -dimensional lattice is unitary, and hence the code is information lossless.

In addition, when asynchronicity between relays is involved, the rank criterion should be also verified for the shifted matrix and another criterion will be analyzed that is the non-zero product distance of the codeword matrix in order to prove that the new codes are delay-tolerant, and thus keep their full-diversity in asynchronous transmission.

## V New Delay-Tolerant Codes from 2,3,4-dimensional Perfect Codes

Based on the previous approach, we consider the perfect codes proposed in [14, 15] for dimensions to construct the new delay-tolerant codes. Then, in the next section, we apply this construction for the perfect codes presented for any number of antennas in [Elia2:2005].

### V-a 2×2 Code based on Golden Code

The Golden Code was constructed in [14] using the cyclic division algebra of degree over . is a Galois extension of degree . It is a -dimensional vector space of with basis , being the Golden number. Its Galois group is generated by . In order to get a rotated lattice of , the principal ideal generated by was found. Its basis is and its unitary generator matrix is given by

 M1=1√5[ααθ1¯α¯α¯θ1], (16)

with and the respective conjugates of and .

Let the cyclotomic extension of degree over with the primitive root of unity. Its discriminant and it is coprime to the one of since . The Galois group is generated by and the integral basis of is . The corresponding unitary generator matrix is

 M2=1√2[1ζ81−ζ8] (17)

Therefore, is the compositum of Galois extensions of degree each, with coprime discriminants. The unitary matrix is obtained by the tensor product of previous matrices as

 M4=1√10⎡⎢ ⎢ ⎢ ⎢⎣ααθ1αζ8αθ1ζ8¯α¯α¯θ1¯αζ8¯α¯θ1ζ8ααθ1−αζ8−αθ1ζ8¯α¯α¯θ1−¯αζ8−¯α¯θ1ζ8⎤⎥ ⎥ ⎥ ⎥⎦ (18)

and the codeword matrix is defined by

 Γ(s)=[ϕ1x1ϕ3x3ϕ2x2ϕ4x4], (19)

where are the components of the vector with are -QAM symbols. We propose now to determine the coefficients that satisfy the non-vanishing determinant criterion.

#### V-A1 Non-vanishing minimum determinant

The determinant of this codeword matrix equals

 δ(s)=ϕ1ϕ4x1x4−ϕ2ϕ3x2x3. (20)

By developing and , we obtain

 x1x4 = 110NK1/F(α)(G(s)+1+i√2√5s1s4−1+i√2√5s2s3), (21) x2x3 = 110NK1/F(α)(G(s)−1+i√2√5s1s4+1+i√2√5s2s3), (22)

with

 NK1/F(α)=α¯α=2+i  and  G(s)=s21−s22−is23+is24+s1s2−is3s4. (23)

It is interesting to note that the Golden codeword given by matrix (7) has a determinant of

 δ′(s)=15NK1/F(α)G(s). (24)

Therefore, by choosing and , the determinant of the new code is equal to the Golden code determinant, and does not vanish when increasing the size of the QAM constellation carved from . Hence, the new code achieves the diversity-multiplexing tradeoff [20, 16].

It can also be noticed that the coefficients can be changed equivalently to the coefficients of the Fourier matrix where is the primitive root of unity. For dimension , we have

 [Φ]=[111−1] (25)

Furthermore, we have find fixed unitary matrices and such that for all values of with

 U=[ζ800−1],V=1√2[−iζ8−iζ81−1]. (26)

#### V-A2 Delay-tolerance

In the distributed setup, each row of the code matrix is transmitted by a different relay (Section III-B). In practical scenarios, the two relays do not share a common timing reference, and therefore, the arrival of packets is not synchronous. As we assume synchronization at the symbol level, the distributed code can still achieve full diversity if the differences between matrix codewords are full rank even when the different rows are arbitrarily shifted. In what follows, we prove that the new code satisfy this condition.

Consider the shifted codeword matrix of

 Γa=[0x1x3−x2x40], (27)

we need to guarantee that it is full rank when i.e., for any from the constellation . This restricts to show that the submatrix

 [0x3−x20]

is full rank i.e., its determinant when .

More generally, having delay profiles or , the problem turns to prove that the product distance in the rotated constellation associated with the matrix of is non-zero over , so that any component product is non-zero. This product distance is evaluated as

 ∣∣110α¯α(G(s)−1+i√2√5s1s4+1+i√2√5s2s3)∣∣=120∣∣ ∣∣G(s)2−(1+i√2√5(s1s4−s2s3))2∣∣ ∣∣ (28)

with for .

As a direct consequence from the tensor product construction, Equation (14) gives

 dp,min=1√dK=1√5242=120

Thus, the minimum product distance is non-zero. It can also be verified in by setting . So, is non-zero unless , and consequently the submatrix is full rank since unless .

Therefore, the new code unlike the Golden code keeps its full-diversity in the case of asynchronous relays. However, we cannot guarantee the non-vanishing determinant property in the asynchronous case because the expression of can be interpreted as a Diophantine approximation of by rational numbers which can be made tighter by using larger constellation size.

### V-B 3×3 Code based on 3×3 Perfect Code

In order to construct the delay-tolerant code, we consider the base field and we use -HEX symbols. Let , with the root of unity. The perfect code was constructed using the cyclic division algebra of order [15], where the relative extension and the generator of the cyclic extension with . The integral basis is given by and the complex lattice is a rotated version of . It is generated by

 M1=1√7⎡⎢⎣v1v2v3σ1(v1)σ1(v2)σ1(v3)σ21(v1)σ21(v2)σ21(v3)⎤⎥⎦. (29)

The relative discriminant of is . Another extension of of degree that has coprime discriminant with is the cyclotomic extension with the primitive root of unity and . Its Galois group is generated by . The integral basis of is and the lattice generator matrix is

 M2=1√3⎡⎢ ⎢⎣1ζ9ζ291jζ9j2ζ291j2ζ9jζ29⎤⎥ ⎥⎦. (30)

The compositum of both extensions is of order over . Then, the corresponding -dimensional complex lattice is generated by the unitary matrix

 M9=1√21⎡⎢ ⎢⎣1ζ9ζ291jζ9j2ζ291j2ζ9jζ29⎤⎥ ⎥⎦⊗⎡⎢⎣v1v2v3σ1(v1)σ1(v2)σ1(v3)σ21(v1)σ21(v2)σ21(v3)⎤⎥⎦, (31)

and the space-time code is defined by the matrix

 Γ(s)=⎡⎢⎣ϕ1x1ϕ4x4ϕ7x7ϕ2x2ϕ5x5ϕ8x8ϕ3x3ϕ6x6ϕ9x9⎤⎥⎦, (32)

where are the components of vector , being the information symbol vector carved from -HEX constellation.

#### V-B1 Non-vanishing minimum determinant

By proceeding as previously, we need to determine the coefficients that guarantee the non-vanishing minimum determinant. In order to get so that a uniform average energy is transmitted per antenna, and to obtain better values of the determinant, we limit the choice of to .

By developing the code determinant using symbolic computation under Mathematica, we find that it has the same expression as the perfect code determinant by choosing as the Fourier matrix coefficients in

 Φ=⎡⎢⎣1111jj21j2j⎤⎥⎦. (33)

Therefore, the infinite code has non-vanishing minimum determinant equal to

 δmin(C)=1dK1=149. (34)

#### V-B2 Delay-tolerance

On the other hand, to prove the delay-tolerance of this code, we should guarantee that the corresponding shifted codeword matrices are full rank. Therefore, it suffices to verify that for each asynchronous matrix there exists a square matrix that is full rank i.e., its determinant is non-zero. In fact, if we enumerate all the delay profiles, it can be noticed that the problem of guaranteeing full-rank shifted matrices turns to guarantee that

1. All component products are non-zero. This condition is always verified since the product distance over as .

2. All minors of are non-zero that is equivalent to verify that the entries of the cofactor matrix of are non-zero.

In order to prove the second condition, we find two unitary matrices and such that the codeword matrix can be written as for all , with is the perfect code matrix and and are defined by

 U=⎡⎢⎣1000j2ζ29000j2ζ9⎤⎥⎦ ,  V=1√3⎡⎢⎣ccc111ζ9jζ9j2ζ9ζ29j2ζ29jζ29⎤⎥⎦. (35)

Let define the cofactor matrix of the perfect code by . Since is a finite subset of the cyclic division algebra , is also a subset of taken from the lattice with and is the ring of integers of . Hence, the cofactor matrix can be represented as a codeword matrix. For simplicity, we denote by and , the conjugates of an entry of the codeword matrix. The cofactor codeword matrix is then defined by

 ˜Z=⎡⎢⎣z1z2z3j¯z3¯z1¯z2j¯¯z2j¯¯z3¯¯z1⎤⎥⎦, (36)

where each diagonal .

Since , we denote its cofactor matrix. It is given by and satisfies

 Γ˜Γ=UZVV†˜ZU†=det(Z)I, (37)

with

 U†=⎡⎢⎣1000ζ9000ζ29⎤⎥⎦ ,  V†=1√3⎡⎢ ⎢⎣1j2ζ29j2ζ91jζ29ζ91ζ29jζ9⎤⎥ ⎥⎦. (38)

Developing the cofactor matrix , we get

 ˜Γ=V†˜ZU† = ⎡⎢ ⎢⎣z1+ζ29¯z3+ζ9¯¯z2ζ9z2+¯z1+ζ29¯¯z3ζ29z3+ζ9¯z2+¯¯z1\parz1+j2ζ29¯z3+jζ9¯¯z2ζ9z2+j2¯z1+jζ29¯¯z3ζ29z3+j2ζ9¯z2+j¯¯z1\parz1+jζ29¯z3+j2ζ9¯¯z2ζ9z2+j¯z1+j2ζ29¯¯z3ζ2