Space-time block codes from nonassociative division algebras

Associative division algebras are a rich source of fully diverse space-time block codes (STBCs). In this paper the systematic construction of fully diverse STBCs from nonassociative algebras is discussed. As examples, families of fully diverse $2\times 2$, $2\times 4$ multiblock and $4\x 4$ STBCs are designed, employing nonassociative quaternion division algebras.


Introduction
Space-time coding is used for reliable high rate transmission over wireless digital channels with multiple antennas at both the transmitter and receiver ends. From a mathematical point of view, a space-time block code (STBC) consists of a family of matrices with complex entries (the codebook) that satisfies a number of properties which determine how well the code performs.
The first aim is to find fully diverse codebooks, where the difference of any two code words has full rank. Once a fully diverse codebook is found it is then further optimized to satisfy additional design criteria (see Section 6).
Most of the existing codes are built from cyclic division algebras over F = Q(i) or F = Q(ζ 3 ) with ζ 3 = e 2πi/3 a third root of unity. These fields are used for the transmission of QAM or HEX constellations, respectively.
There are two ways to embed an associative division algebra into a matrix algebra in order to obtain a codebook: the left regular representation of the algebra and the representation over some maximal subfield. For instance, a real 4 × 4 orthogonal design is obtained by the left regular representation of the real quaternions H = (−1, −1) R (see for instance [30, p. 1458]), whereas the Alamouti code [1] uses the representation of H over its maximal subfield C.
In [29, p. 2608], the authors note that "the Alamouti code is the only rate-one STBC which is full rank over any finite subset of C, which is due to the fact that the set of quaternions H is the only division algebra which has the entire complexes as its maximal subfield." There are however other, nonassociative, division algebras over R of dimension 4 which contain C as a subfield and which yield STBCs that are full rank over any finite subset of C: the nonassociative quaternion division algebras over R which were classified in [4] and which we will employ here.
In this paper we show how nonassociative division algebras can be used to systematically construct fully diverse linear STBCs. If a nonassociative algebra A behaves well enough, one can also obtain fully diverse families of matrices using subfields of A. We use 4-dimensional nonassociative quaternion division algebras to construct new examples of fully diverse 2 × 2 and 4 × 4 space-time block codes and of 2 × 4 multiblock space-time codes (cf. [17]). We also investigate when these codes satisfy the non-vanishing determinant property.
The paper is organized as follows: in Sections 2 and 3 we present nonassociative algebras and the Cayley-Dickson doubling process, respectively. In Section 4 we define nonassociative quaternion division algebras (constructed via a generalized Cayley-Dickson doubling process). In Section 5 we explain the general framework for obtaining fully diverse STBCs from nonassociative division algebras. In Section 6 we list the design criteria used in the construction of STBCs. In Section 7 we look in more detail at the construction of fully diverse 2 × 2 STBCs from nonassociative quaternion algebras. We also discuss the non-vanishing determinant and information lossless properties. This is followed by many examples. In Sections 8 and 9 we discuss the construction of fully diverse 2 × 4 multiblock codes and 4 × 4 codes from nonassociative quaternion algebras, respectively. In Appendix A we collect results from algebraic number theory that are needed in this paper.

Nonassociative algebras
Let F be a field and let A be a finite-dimensional F -vector space. We call A an algebra over F if there exists an F -bilinear map A × A → A, (x, y) → x · y (also denoted simply by juxtaposition xy), called multiplication, on A. This definition does not imply that the algebra is associative; we only have c(xy) = (cx)y = x(cy) for all c ∈ F , x, y ∈ A. Hence we also call such an algebra a nonassociative algebra, in the sense that it is not necessarily associative. A (nonassociative) algebra A is called unital if there is an element in A (which can be shown to be uniquely determined), denoted by 1, such that 1x = x1 = x for all x ∈ A. We will only consider unital nonassociative algebras.
For a nonassociative F -algebra A, associativity in A is measured by the associator The nucleus of A is defined as The nucleus is an associative subalgebra of A (it may be zero), and x(yz) = (xy)z whenever one of the elements x, y, z is in N (A). In other words, the nucleus of the algebra A contains all the elements of A which associate with every other two the middle nucleus of A is defined as and the right nucleus of A is defined as Their intersection is the nucleus N (A). A nonassociative algebra A is called a division algebra if for any a ∈ A, a = 0, the left multiplication with a, λ a (x) = ax, and the right multiplication with a, ρ a (x) = xa, are bijective. The algebra A is a division algebra if and only if A has no zero divisors [25, pp. 15, 16]. Note that if the F -algebra A is associative and finite-dimensional as an F -vector space, this definition of division algebra coincides with the usual one for associative algebras.

The Cayley-Dickson doubling process
The Cayley-Dickson doubling process is a well-known way to construct a new algebra with involution from a given algebra with involution. It can be motivated by the observation that the complex numbers can be viewed as pairs of real numbers with componentwise addition and a suitably defined multiplication: The unit element for this multiplication is (1, 0). Let i = (0, 1). Then i 2 = (−1, 0). We can now write the pair (u, v) as (u, v) = (u, 0) + (0, 1)(v, 0) and identify it with the element u + iv ∈ R ⊕ iR. In this way we obtain the complex numbers Let denote complex conjugation, given by x = u − iv for x = u + iv. Then we can also write (u, v) = (u, −v).
The above process can be repeated with C instead of R: define a multiplication on The unit element for this multiplication is (1, 0). Let j = (0, 1) ∈ C × C. Then j 2 = (−1, 0). We identify the element (u, v) ∈ C × C with u + jv ∈ C ⊕ jC. In this way we obtain the Hamilton quaternions We define quaternion conjugation (again denoted ) via Another iteration of this process, this time starting with H, results in the (Cayley-Graves) octonion algebra O.
Remark 3.2. In 1958 it was shown that finite-dimensional real division algebras can only have dimension 1, 2, 4 or 8 (cf. [10]). In addition to the well-known algebras R, C, H and O, there exist other finite-dimensional real division algebras. The algebras R, C, H and O are just the alternative ones (see [25, p. 48]). Indeed, a complete classification of these algebras is still far away. Only certain subclasses are understood thus far. Moreover, the restriction on the dimension only holds for real and real closed fields (see [11]). Over number fields there exist also higher dimensional division algebras as well as division algebras which do not appear over the real numbers.
The previous construction of the Hamilton quaternions as a double of the complex numbers serves as a motivating example for obtaining generalized (associative) quaternion algebras as doubles, as we will do now.
Let F be a field. Let K be a separable quadratic field extension of F with non-trivial Galois automorphism σ : K → K. Let b ∈ F × := F \ {0}. Then the 4-dimensional F -vector space K × K can be made into a new unital associative (but not commutative) algebra over F via the multiplication The unit element is given by (1,0). The automorphism σ induces an involution on K × K as follows: Let j = (0, 1). Then j 2 = (b, 0). We identify (u, v) ∈ K × K with u + jv in K ⊕ jK. The algebra K ⊕ jK is called the Cayley-Dickson double of K (with scalar b) and denoted by Cay(K, b) (cf. [2]).
The Cayley-Dickson double of K yields a quaternion algebra over F . If F has characteristic not 2 and for every d ∈ F × . The algebra Cay(K, b) is a division algebra if and only if b ∈ N K/F (K × ), where N K/F is the norm of the field extension K/F . The quadratic norm N A : A → F of the algebra A = Cay(K, b) is given by for u, v ∈ K. If F has characteristic not 2, a straightforward computation shows that Example 3.4. The algebra (5, i) Q(i) = Cay(K, i) with K = Q(i, √ 5) is the quaternion algebra used in the construction of the Golden code [21]. This algebra is isomorphic to the cyclic algebra (K/F, σ, i) where σ : is the golden number, we also have that Cay(Q(i, θ), i) ∼ = (5, i) Q(i) .

Nonassociative quaternion division algebras
Let F be a field of characteristic not 2. Let K be a quadratic field extension of F with non-trivial Galois automorphism σ and let b ∈ K \ F . We define an algebra structure on the F -vector space K × K via the multiplication The multiplication is thus defined just as for quaternion algebras with the exception that we require the scalar b to lie outside of F . We denote the algebra again by Cay(K, b). Its unit element is (1, 0).
Since b ∈ K \ F , the multiplication of Cay(K, b) is not associative anymore. It is not even third power-associative, meaning that in general (x 2 )x = x(x 2 ). The algebra Cay(K, b) with b ∈ K and not in F is called a nonassociative quaternion algebra over F . Then A has F -basis {1, i, j, ji} such that i 2 = a, j 2 = b and xj = jσ(x) for all x ∈ K (so in particular ij = −ji).  24] or [34]). The nonassociative quaternion algebra Cay(K, b) has nucleus K and is a division algebra over F .
Thus products involving a factor from K are still associative. Furthermore, nonassociative quaternion algebras are always division algebras, which is not the case for the usual associative quaternion algebras. Over Q, we can easily find non-isomorphic nonassociative quaternion division algebras: it was observed in [34] that two nonassociative quaternion algebras Cay(K, b) and Cay(L, c) can only be isomorphic if L ∼ = K. Moreover, for some automorphism g ∈ Aut(K) and some non-zero d ∈ K.
Nonassociative quaternion algebras provided early examples of real nonassociative division algebras which were neither power-associative nor quadratic. They were investigated for the first time by Dickson [12] in 1935 and by Albert [3] in 1948. In 1987, Waterhouse [34] completely classified these algebras over a field of characteristic not 2.
The only division algebras which appear in the classification of 4-dimensional Kassociative algebras, cf. [4] or [34], are the generalized quaternion division algebras and the nonassociative quaternion division algebras over F .

STBCs from nonassociative division algebras: the general setup
The general setup for constructing a fully diverse STBC from an associative division algebra A is simple: associate to each nonzero element x ∈ A a square matrix X over a fixed subfield of A (normally the base field, via the left regular representation, or a maximal extension of the base field). The difference of any two such matrices X − X ′ (with X = X ′ ) will then always be invertible. This procedure can be adapted to work in the nonassociative case as well, as will be explained in this section.
5.1. The left regular representation. Let A be a nonassociative division algebra over F of dimension n as an F -vector space. Let a be any element in A. The left multiplication λ a : A → A determined by a is defined by x → ax for all x ∈ A. The operator λ a is linear and the set {λ a | a ∈ A} is a subspace of the associative algebra End F (A), the algebra of F -linear transformations on A. Consider the left regular representation λ : If λ a = λ b then ax = bx for all x ∈ A, hence (a − b)x = 0 for all x, which yields a = b (A is a finite-dimensional division algebra and as such does not have zero divisors) and we have an injection. After a choice of F -basis for A, we can embed End F (A) into the algebra Mat n (F ) of n × n-matrices with entries from F , where n = dim F (A). In this way we get an embedding λ : A ֒→ Mat n (F ) of vector spaces.
Contrary to the situation for associative division algebras, this only embeds the vector space A into the vector space Mat n (F ); the algebra structure of A is disregarded here.
Nonetheless, all non-zero elements of A are invertible, hence all λ a with a = 0 are bijective and so all non-zero matrices in λ(A) have non-zero determinant. Now X ± Y ∈ λ(A) for all X, Y ∈ λ(A). Thus λ(A) constitutes a linear codebook which in addition is fully diverse, since the rank of the difference of two distinct codewords is maximal.

5.2.
Representation over a maximal subfield. For coding purposes, an associative division algebra A is often considered as a vector space over some subfield K of the algebra A. Usually K is maximal with respect to inclusion. Given a nonassociative F -algebra A with a maximal subfield K, this is not always possible because of the nonexistence of the associative law. So what are the minimum requirements on a nonassociative algebra in order to have such a representation?
Let K be a subfield of the F -algebra A. We need A to be a right K-vector space, i.e. we need We also need that left multiplication λ a is a linear endomorphism of the right K- for all a, x ∈ A, α ∈ K and λ a ∈ End K (A), so λ : Thus, let K be a subfield of A, maximal with respect to inclusion and assume . Consider A as a right K-vector space. After a choice of K-basis for A, we can embed End K (A) into the vector space Mat r (K) where r = dim K (A). In this way we get an embedding λ : A ֒→ Mat r (K) of vector spaces. Obviously, we have X ± Y ∈ λ(A) for all X, Y ∈ λ(A). Thus λ(A) constitutes a linear codebook.
Remark 5.1. If we want to consider A as a left K-vector space, we require K ⊂ N ℓ (A) and K ⊂ N m (A) and adjust the above construction accordingly.
More generally one can do the following: let D be a subalgebra of A, assume that and suppose A can be viewed as a free right D-module of rank r. After a choice of a D-basis for A, we can embed the right D-module End D (A) into the vector space Mat r (D). In this way we get an embedding λ : Remark 5.2. It is not known whether there exist 8-dimensional real division algebras with some (left, middle or right) nucleus isomorphic to H. The fact that there are no 8-dimensional real division algebras with two associative nuclei (left, middle or right) isomorphic to H suggests a negative answer [16,Proposition 3]. This need not be the case over other base fields, however.
In the remainder of this paper, all fields are assumed to be algebraic number fields unless stated otherwise.

STBC Design criteria
Let C ⊂ Mat n (C) be a space-time block code. In order for C to perform well, it should satisfy property (1) below (as remarked before) and as many of the other properties as possible.
(2) It has full rate, which means that the n 2 degrees of freedom are used to transmit n 2 information symbols. (3) It has non-vanishing determinant : the minimum determinant of the code, is bounded below by a constant even if the codebook C is infinite. (4) It has cubic shaping: each layer of a codeword is of the form Rv, where R is a unitary matrix and v is a vector containing the information symbols. As a consequence it is information lossless. (5) It induces uniform average energy per antenna: the ith antenna will transmit the ith row of the codeword; on average, the norms of the rows should be equal in order to have a balanced repartition of the energy at the transmitter. These properties, originally considered for codes based on associative division algebras, also make sense in the nonassociative case. Codes that satisfy all the properties above are called perfect codes. We refer to [22] for more details. The Golden Code [6] is the best performing 2 × 2 perfect STBC, cf. [21].

2 × 2 Codebooks from nonassociative quaternion division algebras
STBCs based on associative quaternion algebras seem to have been considered explicitly for the first time in [5]. See also [32] for more details. In this section we look at the construction of STBCs based on nonassociative quaternion algebras.
Roughly speaking, constructing a nonassociative quaternion division algebra boils down to choosing the nonzero scalar b in the quadratic field extension K = F ( √ a) of the base field F , and not in F itself. In contrast, b is chosen in F in the construction of a classical generalized quaternion algebra (a, b) F over F . This usually gives us more freedom of choice for b, despite the restriction that we will still have to require |b| 2 = 1 in order to get a balanced repartition of the energy at the transmitter. The choice of F = Q(i) allows us to transmit QAM constellations. 7.1. Fully diverse codebook construction. Let K be a quadratic field extension of F with non-trivial Galois automorphism σ.
The algebra A is K-associative by Theorem 4.2, hence we can consider A as a right vector space over the subfield K of A. The field K is maximal with respect to inclusion. For x ∈ A, the left multiplication λ x : A → A, a → xa, is a K-linear endomorphism of the right K-vector space A. Therefore λ x ∈ End K (A) and we get an injective K-linear map Consider the K-basis {1, j} of A. Then End K (A) ∼ = Mat 2 (K) as vector spaces and we get an embedding λ : A ֒→ Mat 2 (K) of vector spaces, which sends x ∈ A to the matrix of λ x with respect to the basis {1, j}. Then by the rules in Remark 4.1.

Lemma 7.2. For any
, the difference of any two distinct elements in λ(A) will have non-zero determinant. Therefore the (infinite linear) codebook built on A, C := λ(A), is fully diverse. 7.2. Non-vanishing determinant. We closely follow the approach in [8, §17]. The minimum determinant of C determines the coding gain and is defined as The discussion in [8, p. 73] can easily be adapted to the more general set-up of nonassociative algebras. Since the codebook C is linear (it is based on an algebra) we have Let us compute the minimum determinant of the codebook with the infimum taken over all (c, d, e, f ) = (0, 0, 0, 0), or equivalently Thus δ(C) ∈ K ∩ R + . Since A is a division algebra, δ(C) = 0. If the code C is finite, i.e. if the information symbols c, d, e, f belong to a finite constellation in F , then δ(C) is bounded below by a constant. If the constellation size increases however, δ(C) can get arbitrarily close to zero (e.g. let (c, d, e, f ) = ( 1 n , 0, 0, 0); as n increases, δ(C) will approach zero). This will also be the case for infinite codes.
Codes whose minimum determinant is bounded below by a constant which is independent of the size of the constellation from which the information symbols are chosen are said to satisfy the non-vanishing determinant (NVD) property, cf. Section 6.
For associative division algebras over a number field F and with maximal subfield K infinite codes that satisfy the NVD property can often be obtained by restricting the entries in the codebook to the ring of integers O K . If F = Q or F is quadratic imaginary, then the resulting code will still be infinite, and its minimum determinant is guaranteed to be bounded away from zero, cf. [8,Cor. 17.8].
Let us look at what happens for a code C, based on a nonassociative quaternion division algebra.
If K is quadratic imaginary, then there exists an integer d > 0 such that δ(C OK ) ≥ 1 d (and so C OK satisfies the NVD property), otherwise δ(C OK ) can become arbitrarily small.
with the infimum taken over all (u, v) = (0, 0). Taking c = |b d | 2 establishes the first part of the proposition.
Assume that K is quadratic imaginary (i.e. F = Q and a < 0). Then it follows from Proposition A.1 that O K ∩ R + = N and that |b d If K is not quadratic imaginary it follows from the Dirichlet Unit Theorem (Proposition A.8) that O K contains units u such that |u| 2 is arbitrarily large or small, so that δ(C OK ) can become arbitrarily small. Remark 7.4. It follows from the proposition that codes based on nonassociative quaternion algebras only satisfy the NVD property if we assume that K is a quadratic imaginary number field.
Assume that K is quadratic imaginary. If we assume in addition that O K is a unique factorization domain (or, equivalently, a principal ideal domain; cf. Appendix A), we can write b as an irreducible fraction b = b n /b d and b d will be unique up to multiplication by a unit. By the Dirichlet Unit Theorem the only units in O K are roots of unity, so that |b d | 2 is unique. For information symbols u, v taken from a QAM constellation we use the field K = Q(i) which is quadratic imaginary and whose ring of integers O K = Z[i] is a unique factorization domain. 7.3. Information lossless encoding. A code C is information lossless if it is obtained from information symbols in such a way that the energy needed to transmit them is the same as the energy needed to transmit the information symbols without encoding. By [23,Prop. 3.5] it suffices to construct the layers of each codeword from the information symbols vector by applying a unitary matrix. This procedure is called cubic shaping (cf. [22, p. 3886]), as it corresponds to an isometry transformation of the cubic lattice Z[i] n in the case of information symbols taken from a QAM constellation. The term good shaping is also used.
The energy needed to transmit a complex number z is determined by |z| 2 . The energy needed to transmit a codeword X = [x i,j ] ∈ C is determined by its squared Frobenius norm X 2 = i,j |x i,j | 2 .
Information lossless encoding of information symbols c, d, e, f ∈ F into a codeword of C can be done as follows. Let {u 0 , u 1 } be an F -basis of K. Let be the matrix of the embeddings of the basis. Let x 0 = cu 0 +du 1 and x 1 = eu 0 +f u 1 be elements of K and consider the vectors x 0 = (c, d) T , x 1 = (e, f ) T , each containing two information symbols. Then Gx 0 = (x 0 , σ(x 0 )) T , Gx 1 = (x 1 , σ(x 1 )) T .
Let Γ 1 = I 2 be the identity matrix and Then we can write X as the sum of its layers, In order for the encoding to be information lossless, we want Γ 2 and G to be unitary. Note that the matrix Γ 2 is unitary if and only if |b| 2 = 1.
In order to find a good unitary matrix G (if one exists) and also to satisfy the NVD property, one usually restricts x 0 and x 1 to the ring of integers O K or an ideal I of O K with "good" properties. See [23, §4.3-4.4] for more details.
In the rest of this section we construct fully diverse 2 × 2 codebooks λ(A), based on nonassociative quaternion algebras A. From these we construct codebooks C that satisfy the NVD and/or cubic shaping properties in certain cases. The codebooks are all infinite. When restricting entries of the codewords to the ring of integers O K , we indicate this by writing C OK .
The codebook obtained from Cay(C, i) closely resembles the Alamouti Code. Consider also B = Cay(C, −i). Then resulting in a shaped code C B with codewords In examples (ii) and (iii), F = Q, K = Q(i) is a quadratic imaginary number field, b = ±i and O K = Z[i] is a principal ideal domain. Hence, before shaping, the minimum determinant of each code is bounded below by 1 by Proposition 7.3. Thus the minimum determinant of both shaped codes is lower bounded by the constant 1/2.
To summarize: the codes in (ii) and (iii) are fully diverse, satisfy the NVD property, have good shaping and clearly also satisfy the uniform average transmitted energy per antenna property. Used for a 2 × 2 MIMO channel they are only halfrate though, since 4 transmitted signals are used to transmit 2 QAM information symbols.
To obtain an energy-efficient code, the entries in the codewords are then restricted to elements in the principal ideal I in O K of norm 5 in Q which is generated by α = 1 + i − iθ. So, finally the Golden Code is given by the codewords with c, d, e, f ∈ Z[i] and has minimum determinant 1/5. We refer to [6] for the details.
Example 7.6. Consider the nonassociative quaternion division algebra The codebook based on A is (Compared to the general code construction in Lemma 7.1, we are transposing the matrices here in order to better compare them with the Golden Code matrices above. This does not influence the behaviour of the code.) The code has full diversity and uniform average transmitted energy per antenna. Now {1, θ} is a Q(i)-basis of Q(i)( √ 5), but is not a unitary matrix, so we have an energy shaping loss. To obtain an energyefficient code, restrict the entries in the codeword again to elements in the principal ideal I in O K generated by α = 1 + i − iθ. Then a nonassociative Golden Code is given by the codewords The choice of the ideal I is optimal here for the exact same reasons as the ones given in [6, p. 1433] and yields good shaping. The code is also full rate, fully diverse and has uniform average transmitted energy per antenna.
With the same arguments codes can be constructed using any of the infinitely many scalars b ∈ Q(i)( √ 5) \ Q(i) with |b| 2 = 1. All these codes, however, have vanishing determinant by Proposition 7.3. We give another example: and consider the nonassociative quaternion algebra Cay Q(i)( √ 5), 2i + √ 5 3 over Q(i). Again | 2i+ √ 5 3 | 2 = 1 and we obtain another code which has full diversity and uniform average transmitted energy per antenna. To obtain an energy-efficient code, we restrict the entries in the codeword again to elements in the principal ideal I in O K generated by α = 1 + i − iθ. Then another nonassociative Golden Code with good shaping is given by the codewords 7.6. Optimality of the Golden Code. Oggier [21] shows that the Golden Code is optimal inside the class of cyclic algebra based 2 × 2 codes built over fields K = Q(i)( √ d) in the following sense: the minimum determinant of such codes is inversely proportional to |d K/Q(i) |, where d K/Q(i) denotes the relative discriminant of K/Q(i). For the Golden Code |d K/Q(i) | = 5. While it is possible to consider fields K = Q(i)( √ d) with |d K/Q(i) | < 5, Oggier shows that the resulting codes are no longer fully diverse [21,III].
This problem does not occur in the nonassociative case by Theorem 4.2. It is possible to construct fully diverse nonassociative codes over fields K = Q(i)( √ d) with |d K/Q(i) | < 5, but by Proposition 7.3 these codes do not satisfy the NVD property.
In the examples below we will consider the cases |d K/Q(i) | = 4 and |d K/Q(i) | = 3. The case |d K/Q(i) | = 2 does not exist, cf. Proposition A.5.
2 is an 8th root of unity and σ(ζ 8 ) = −ζ 8 . We Consider the nonassociative quaternion division algebra A = Cay(Q(ζ 8 ), ζ 8 ) over Q(i). The choice of b = ζ 8 guarantees that Γ 2 is unitary, since |ζ 8 | 2 = 1. We obtain the codebook is a unitary matrix. So after multiplying the matrices in the codebook by 1 √ 2 and restricting the information symbols to Z[i], we obtain a code that has good shaping: The code C is full rate, has full diversity and good shaping. The factor ζ 8 in the first row of the codeword guarantees uniform average transmitted energy per antenna since |ζ 8 | 2 = 1. This code does not satisfy the NVD property however by Proposition 7.3.
Example 7.9. Let K = Q(i)( √ 3). Then |d K/Q(i) | = 3 and σ( is a third root of unity. We have Consider the nonassociative quaternion division algebra A = Cay(Q(i)(ζ 3 ), ζ 3 ) over Q(i). We obtain the codebook This time the matrix G (up to scaling) is not unitary. Thus the energy required to send the linear combination of the information symbols on each layer is higher than the energy needed to send the information symbols themselves and we would still have to optimize for energy efficiency. In addition the discriminant of this code is not bounded away from zero by Proposition 7.3.

2 × 4 Multiblock space-time codes from nonassociative quaternion algebras
Let F be a number field, let a ∈ F × and let K = F ( √ a) be a quadratic field extension of F with non-trivial Galois automorphism σ and norm N K/F (x) = xσ(x). Let b ∈ K \ F , so that A = Cay(K, b) is a nonassociative quaternion division algebra. A 2 × 4 multiblock space-time code based on A is a set of matrices of the form Y = [X|σ(X)] where X ∈ λ(A) (see [17] for a more general construction in the associative case).
This yields a codebook C consisting of matrices of the form They have full rank since X comes from the division algebra A.
In this set-up, we want the code to satisfy a generalized NVD property (see [18]) which can be achieved by bounding the generalized minimum determinant | det(X) det(σ(X))| away from zero.
If F = Q or F is quadratic imaginary, then there exists an integer d > 0 such that δ g (C OK ) ≥ 1 √ d (and so C OK satisfies the generalized NVD property), otherwise δ g (C OK ) can become arbitrarily small.
Proof. Write b as a fraction b = bn b d with b n , b d ∈ O K (not necessarily unique) and b d = 0. We have Next assume that F is quadratic imaginary (i.e. F = Q( √ m) and m < 0). Then it follows from Proposition A.1 that O F ∩ R + = N and that If F is not Q or not quadratic imaginary it follows from the Dirichlet Unit Theorem (Proposition A.8) that O F contains units u such that |u| 2 is arbitrarily large or small, so that δ g (C OK ) can become arbitrarily small. Remark 8.2. If F = Q or F is quadratic imaginary, the generalized minimum determinant of C OK is lower bounded by a positive constant and the generalized NVD property is satisfied. As a consequence the code will achieve the diversitymultiplexing gain trade-off, as explained in [17, p. 5232].
If we assume in addition that O K is a unique factorization domain (or, equivalently, a principal ideal domain; cf. Appendix A), we can write b as an irreducible fraction b = b n /b d and b d will be unique up to multiplication by a unit. By the Dirichlet Unit Theorem the only units in O K are roots of unity, so that For QAM constellations we take F = Q(i), so that K = Q(i)( √ m) for some square-free non-zero integer m. Proposition A.6 lists the fields K whose ring of integers O K is a unique factorization domain.
In the setting of multiblock space-time codes it is again natural to ask that |b| 2 = 1, cf. [17, p. 5232]. be the golden number, σ : over Q(i) as in Example 7.6. In order to obtain an energy-efficient code, we restrict the entries in the codewords to elements in the principal ideal I in O K , generated by α = 1 + i − iθ. Then with c, d, e, f ∈ Z[i] and the code C OK consists of block matrices of the form Note that guaranteeing that the code satisfies the generalized NVD property. (cf. Example 7.7) in the previous example results in a code C OK such that guaranteeing that the code satisfies the generalized NVD property.
is an 8th root of unity as in Example 7.8. Let A = Cay(Q(ζ 8 ), ζ 8 ). Then and the code C OK consists of block matrices of the form We have guaranteeing that the code satisfies the generalized NVD property.
Let us look at the left regular representation of a nonassociative quaternion algebra A, this time over its base field rather than over a maximal subfield. 9.1. Fully diverse codebook construction. Let F be a number field and let K = F ( √ a) = F (i) with i 2 = a ∈ F × be a quadratic field extension with nontrivial Galois automorphism σ : √ a → − √ a. Let A = Cay(K, b) be a nonassociative quaternion division algebra over F with b = p + qi ∈ K \ F , so p, q ∈ F with q = 0. For the basis {1, i, j, −ij} of A over F the matrix representation of left multiplication with x = x 0 + x 1 i + x 2 j − x 3 ij yields the fully diverse 4 × 4 space-time block code Its matrices are not orthogonal, but their first two column vectors and, respectively, their last two, are orthogonal to each other. 9.2. Non-vanishing determinant. Let X ∈ C. Then Since the codebook is based on a division algebra, its minimum determinant equals and is non-zero. If the information symbols x 0 , x 1 , x 2 , x 3 belong to a finite constellation in F , then δ(C) is bounded below by a constant which depends on the constellation size. If the constellation size increases, δ(C) can get arbitrarily close to zero. By restricting the entries in C to the ring of integers O F we obtain for certain number fields F infinite codes that satisfy the NVD property: Proposition 9.2. Let F be a number field and let K = F ( √ a) for some nonzero square-free a ∈ O F . Let b = p + q √ a ∈ K \ F with p, q ∈ F (so that q = 0). Let C = λ(Cay(K, b)) and let C OF denote the code obtained from C by restricting the elements of F to elements of O F . Then there exists a constant c > 0 such that If F = Q or F is quadratic imaginary, then there exists an integer d > 0 such that δ(C OF ) ≥ 1 d (and so C OF satisfies the NVD property), otherwise δ(C OF ) can become arbitrarily small.
Proof. Write p = pn p d , q = qn q d with p n , p d , q n , q d ∈ O F (not necessarily unique) and p d = 0, q d = 0. An easy calculation confirms that Letting c = |q d | 4 if p = 0 and c = |p d q d | 4 if p = 0 establishes the first part of the proposition.
Assume for the sake of argument that p = 0. The case p = 0 can be settled in a similar manner.
Assume that F = Q. Then |q d | 4 is a positive integer. Thus, among all possible pairs (q n , q d ) ∈ O 2 F = Z 2 (with q d = 0) that satisfy q = qn q d , we can choose a pair (q n , q d ) in such a way that |q d | 4 is minimal. Furthermore, O F ∩ R + = N. We let d = |q d | 4 in this case.
Next assume that F is quadratic imaginary (i.e. F = Q( √ m) and m < 0). Then it follows from Proposition A.1 that O F ∩ R + = N and that |q d | 4 = N F/Q (q d ) 2 is a positive integer. Thus, among all possible pairs (q n , q d ) ∈ O 2 F (with q d = 0) that satisfy q = qn q d , we can choose a pair (q n , q d ) in such a way that |q d | 4 is minimal.
If F is not Q or not quadratic imaginary it follows from the Dirichlet Unit Theorem (Proposition A.8) that O F contains units u such that |u| 2 is arbitrarily large or small, so that δ(C OF ) can become arbitrarily small. Remark 9.3. If F = Q or F is quadratic imaginary, the minimum determinant of C OF is lower bounded by a positive constant and the NVD property is satisfied.
If we assume in addition that O F is a unique factorization domain (or, equivalently, a principal ideal domain; cf. Appendix A), we can write p and q as irreducible fractions p = pn p d , q = qn q d and p d , q d will be unique up to multiplication by a unit. By the Dirichlet Unit Theorem the only units in O F are roots of unity, so that |q d | 4 , resp. |p d q d | 4 , is unique.
Example 9.4. The Q-algebra A = Cay(Q(i), i) yields the fully diverse 4 × 4 spacetime block code We obtain Example 9.5. The Q-algebra A = Cay(Q(i), −i) yields the fully diverse 4 × 4 space-time block code again with minimum determinant 1.
Thus, C is information lossless if |a| 2 = |p| 2 + |q| 2 = 1 and pq + aqp = 0. It is not difficult to verify that the codes in Examples 9.4, 9.5 and 9.6 are all information lossless.

Appendix A. Facts from Number Theory
In this appendix we collect some results from algebraic number theory for the convenience of the reader.
Let K be a number field. The ring of integers O K of K is a Dedekind domain [20, I(3.1)].
Let d K denote the discriminant of K. For an extension of number fields K/F , let d K/F denote the relative discriminant of K over F .