Natural orders for asymmetric space–time coding: minimizing the discriminant Amaro Barreal · Capi Corrales Rodrig´ an ˜ ez · Camilla Hollanti
arXiv:1701.00915v1 [math.NT] 4 Jan 2017
the date of receipt and acceptance should be inserted later
Abstract Algebraic space–time coding — a powerful technique developed in the context of Multiple-Input Multiple Output (MIMO) wireless communications — can only be expected to realize its full potential with the help of Class Field Theory and, more concretely, the theory of central simple algebras and their orders. During the last decade, the study of space–time codes for practical applications, and more recently for future generation (5G+) wireless systems, has provided a practical motivation for the consideration of many interesting mathematical problems. One of them is the explicit computation of orders of central simple algebras with small discriminants. We will consider the most interesting asymmetric MIMO channel setups and, for each of these cases, we will provide explicit pairs of fields giving rise to a cyclic division algebra whose natural order has the minimum possible discriminant. Keywords Central Simple Algebras · Division Algebras · Minimum Discriminant · MIMO · Multiple-Access Channel · Natural Orders · Space– Time Coding.
1 Introduction The existing contemporary communications systems can be abstractly characterized by the conceptual seven-layer Open Systems Interconnection Model. The lowest (or first) layer, known as the physical layer, aims to describe the A. Barreal Department of Mathematics and Systems Analysis, Aalto University, Finland Tel.: +358-50-5935194 E-mail:
[email protected] C. Corrales Rodrig´ an ˜ez Faculty of Mathematical Sciences, Complutense University of Madrid, Spain C. Hollanti Department of Mathematics and Systems Analysis, Aalto University, Finland
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communication process over an actual physical medium. Due to the increasing demand for flexibility, information exchange nowadays often occurs via antennas at the transmitting and receiving end of a wireless medium, e.g., using mobile phones or tablets for data transmission and reception. An electromagnetic signal transmitted over a wireless channel is prone to interference, fading, and environmental effects caused by, e.g., surrounding buildings, trees, and vehicles, making reliable wireless communications a challenging technological problem. With the advances in communications engineering, it was soon noticed that increasing the number of spatially separated antennas at both ends of a wireless channel, as well as adding redundancy by repeatedly transmitting the same information encoded over multiple time instances1 , can dramatically improve the transmission quality. A code representing both diversity over time and space is thus called a space–time code. Let us assume nt and nr antennas at the transmitting and receiving end of the channel respectively, as well as T consecutive time instances for transmission. If nt = nr , the channel is called symmetric, and otherwise asymmetric, which more precisely typically refers to the case nr < nt . For the time being, a space–time code X will just be a finite collection of complex matrices in Mat(nt × T, C). The well known channel equation in this Multiple-Input Multiple-Output (MIMO) setting takes the form Ynr ×T = Hnr ×nt Xnt ×T + Nnr ×T , (1) where Y and X are the received and transmitted codeword matrices, respectively. In the above equation, fading is usually modeled as a Rayleigh distributed random process and represented by the random complex channel matrix H, and additive noise2 is modeled by the noise matrix N , whose entries are independent, identically distributed complex Gaussian random variables with zero mean. The main object in (1) is the space–time code matrix X = (xij ), a complex matrix which captures the data to be transmitted across multiple antennas. Namely, the complex number xij is transmitted from the ith antenna in the j th channel use. Let us briefly discuss what constitutes a ”good” code. Consider a space– time code X , and let X, X 0 be code matrices ranging over X . Two basic design criteria can be derived in order to minimize the probability of error [1]. – Diversity gain criterion: To be able to distinguish between two different codewords, we should first maximize the minimum rank of the difference of pairwise distinct matrices X − X 0 . A space–time code achieving the maximal minimum rank min rank(X − X 0 ) = min{nt , T }
X6=X 0
is called a full-diversity code. 1 2
‘Time instances’ are commonly referred to as channel uses. The noise is a combination of thermal noise and noise caused by the signal impulse.
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– Coding gain criterion: If we assume a full-diversity code, then the pairwise decoding error probability Pe = P (X → X 0 ) of a codeword X being confused with another one X 0 , i.e., X 0 6= X is recovered when X was transmitted, can be asymptotically upper bounded by nr nt nr −1 κ Pe ≤ det (X − X 0 )(X − X 0 )† , where κ describes the quality of the channel3 . Thus, ∆min (X ) := min 0 det X − X 0 )(X − X 0 )† X6=X
should be as big as possible.
If we let the size of the code grow, |X | → ∞, and still have inf ∆min (X ) > 0, the space–time code is said to have the nonvanishing determinant property [2]. In other words, a nonvanishing determinant guarantees that the minimum determinant is bounded from below by a positive constant even in the limit, and hence the error probability will not blow up when increasing the code size. Consequently, a good space–time code should ideally be composed of fullrank matrices with large, nonvanishing minimum determinants. ”Large” here is a relative notion and to make it sensible we need some kind of a normalization. We will come back to this in Section 2.1. In 2003, the usefulness of central simple algebras to construct space–time codes meeting both of the above criteria was established in [3]; especially (cyclic) division algebras, for which the property of being division immediately implies full diversity. Thereupon the construction of space–time codes started to rely on cleverly designed algebraic structures. The initially considered cyclic division algebras were however constructed using transcendental elements, which in turn resulted in codes with vanishing minimum determinants. Later on, it was shown in [2] that in a cyclic division algebra based code, the nonvanishing determinant property is enough to achieve the optimal trade-off between diversity and multiplexing; it was also proved that achieving the nonvanishing determinant property of a cyclic-division-algebra-based code can be ensured by restricting the entries of the codewords to certain subrings of the algebra alongside with a smart choice for the base field. Further investigation carried out in [4, 5] showed that codes constructed from orders, in particular maximal orders, of cyclic division algebras actually outperform codes that had been considered unbeatable. The improvement in performance, however, came with the price of somewhat higher complexity in encoding and decoding, as well as more problematic bit labeling of the codewords. As is well known, a maximal order of a cyclic division algebra is not necessarily unique. As a consequence, carrying out the explicit computations required for the purpose of space–time coding is a very challenging task. For this 3 The parameter κ is related to the so-called signal-to-noise ratio (SNR), for more information see [1]. ‘Asymptotically’ above means that we assume κ is relatively large, that is, the signal is of good quality. This is a standard assumption in code design.
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reason, and as a compromise for reducing the complexity of communication while still guaranteeing good performance, the use of natural orders — the main objects of our work — is often preferred instead. However, the current explicit constructions are typically limited to the symmetric case, while the asymmetric case remains largely open. The main goal of this paper is to fill this gap in the construction of explicit optimal (with certain given assumptions, in our case the order being natural) asymmetric space–time codes. In Section 2 we will shortly introduce MIMO space–time coding and the construction of space–time codes using representations of orders in central simple algebras. Section 3 contains the main results of this article. We will consider the most interesting asymmetric MIMO channel setups and fix F = Q or F = Q(i) as the base field to guarantee the nonvanishing determinant property4 . For each such setup (F, nt , nr ), we will find an explicit field extension F ⊂ L ⊂ E and an explicit L-central cyclic division algebra with coefficients in E, such that the norm of the discriminant of its natural order is minimal. This will translate into the largest possible determinant (see [5] for the proof) and thus provide us with the maximal coding gain one can achieve by using a natural order. 2 Space–time codes from orders in central simple algebras From now on, and for the sake of simplicity, we set the number nt of transmitting antennas equal to the number T of time slots used for transmission. Thus, the considered codewords will be square matrices. 2.1 Space–time lattice codes Very simplistically defined, a space–time code is a finite set of complex matrices. However, in order to avoid accumulation points at the receiver, in practical implementations it is convenient to impose an additional discrete structure on the code, such as a lattice structure. Consequently, we define a space–time code to be a finite subset of a lattice Λ in Mat(n, C). We recall that a full lattice Λ ⊂ Mat(n, C) is a lattice with rank(Λ) = 2n2 . We call a space-time lattice code symmetric, if its underlying lattice is full. Otherwise it is called asymmetric 5 . It is not difficult to see that, given a lattice Λ ⊂ Mat(n, C) and X, X 0 ∈ Λ, 2 ∆min (Λ) := inf 0 det (X − X 0 )(X − X 0 )† = inf |det(X)| . X6=X
06=X∈Λ
4 From a mathematical point of view, any imaginary quadratic number field would give a nonvanishing determinant, but the choice Q(i) matches with the quadrature amplitude modulation (QAM) commonly used in engineering. 5 This definition relates to the fact that a symmetric code carries the maximum amount of information (=dimensions) that can be transmitted over a symmetric channel without causing accumulation points at the receiving end. In an asymmetric channel, a symmetric code will result in accumulation points, and hence asymmetric codes, i.e., non-full lattices are called for. See [6] for more details.
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This implies that any lattice Λ satisfying the nonvanishing determinant property can be scaled so that ∆min (Λ) achieves any wanted nonzero value. Consequently, in order to be able to compare different lattices for the purpose of space–time coding, we will need some kind of a normalization. To this end, let B1 , . . . , B2n2 form a basis of a full lattice Λ with volume ν(Λ), and consider its Gram matrix h i GΛ := < Bi Bj† . 2 1≤i,j≤2n
We have det(GΛ ) = ν(Λ)2 .
i) The normalized minimum determinant [5] of Λ is the minimum determinant of Λ after scaling it to have a unit size fundamental parallelotope, that is, ∆min (Λ) . δ(Λ) = 1 ν(Λ) 2n ii) The normalized density [5] of Λ is µ(Λ) =
∆min (Λ)2n . ν(Λ) 1
We get the immediate relation δ(Λ)2 = µ(Λ) n , guaranteeing that in order to maximize the coding gain it suffices to maximize the density of the lattice. Maximizing the density, for its part, translates into a certain discriminant minimization problem [5]. This observation is crucial and will be the main motivation behind our work in Section 3. 2.2 Central simple algebras and orders We aim at constructing space–time codes from lattices within central simple division algebras defined over number fields. We recall that a finite dimensional algebra over a number field L is an L-central simple algebra, if its center is precisely L and it has no nontrivial ideals. An algebra is said to be division if all of its nonzero elements have a multiplicative inverse. As we shall soon see, as long as the underlying algebraic structure of a space–time code is a division algebra, the full-diversity property of the code will be guaranteed. It turns out that if L is an algebraic number field, then every L-central simple algebra is of a certain special type known as cyclic algebras [7, Thm. 32.20]. Throughout the paper, we will denote the relative field norm map of E/L by N mE/L and the absolute norm map shortly by N mE = N mE/Q . When considering orders, we may specify this by writing N mOE /OL although the map naturally remains the same. Let E/L be a cyclic extension with the Galois group Gal(E/L) = hσi. We fix an element γ ∈ L× = L \ {0} and consider the L-central simple algebra as a right E-vector space C := (E/L, σ, γ) =
n−1 M i=0
ui E,
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with left multiplication defined by xu = uσ(x) for all x ∈ E, and un = γ. The algebra C is referred to as a cyclic algebra of index n. A necessary and sufficient condition6 [5, Prop. 3.5] for a cyclic algebra (E/L, σ, γ) of index n to be division is to have γ n/p 6∈ NmE/L (E × ) = {xσ(x) · · · σ n−1 (x) | x ∈ E × } for all prime factors p of n. Consequently, we refer to an element γ satisfying this condition as a non-norm element. The obvious choice of lattices in C will be its orders. We recall that if R ⊂ L is a Dedekind ring, an R-order in C is a subring O ⊂ C which shares the same identity as C, is a finitely generated R-module, and generates C as a linear space over L. Of special interest is the natural order of C, the OL -module Onat :=
n−1 M i=0
ui OE .
If the element γ fails to be an algebraic integer, then Onat will not be closed under multiplication. Consequently, we will always choose γ ∈ L× such that × γ ∈ OL ,
γ n/p 6∈ NmE/L (E × ) for all primes p | n.
(2)
Remark 1 While the ring of algebraic integers is the unique maximal order in an algebraic number field, an L-central division algebra may contain several maximal orders. They all share the same discriminant over OL , known as the discriminant dC of the algebra C. We recall that given a Dedekind ring R ⊂ L and an R-order O with basis {x1 , .., xn } over R, the R-discriminant of O is the ideal in R generated by disc(O/R) = det trC/L (xi xj )ni,j=1 , where tr(·) denotes the reduced trace (cf. (5)). Given two OL -orders Γ1 , Γ2 , it is clear that if Γ1 ⊆ Γ2 , then disc(Γ2 /OL ) divides disc(Γ1 /OL ). Consequently, dC | disc(Γ2 /OL ) for every OL -order Γ2 in C, and the ideal norm NmOL (dC ) is the smallest possible among all OL -orders of C.
The constructions that we will derive in Section 3 will rely on some key properties of cyclic division algebras and their orders that we will next present as lemmata. Lemma 1 [7, Thm. 10.1] Let O be any order in a cyclic division algebra (E/L, σ, γ). Then, for any nonzero element c ∈ O, its reduced norm nr(c) and reduced trace tr(c) (cf. (5)) are nonzero elements of the ring of integers OL . 6 The condition given here is a straightforward generalization and simplification of the original result due to A. Albert, which states that a cyclic algebra is division, if the smallest power of γ that is a norm in E/L is γ n .
Natural orders for asymmetric space–time coding: minimizing the discriminant
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Lemma 2 [5, Lem. 5.4] Let (E/L, σ, γ) be a cyclic division algebra of index × n with a non-norm element γ ∈ OL . We have disc(Onat /OL ) = disc(E/L)n · n(n−1) γ . Hence, if F ⊂ L, then disc(Onat /OF ) = NmL/F (disc(Onat /OL )) · disc(L/F )n n(n−1)
= disc(E/F )n · NmL/F
(γ).
2
(3)
Lemma 3 [5, Thm. 6.12] Assume that L is a number field and that p1 and p2 are a pair of norm-wise smallest prime ideals in OL . If we do not allow ramification on infinite primes, then the smallest possible discriminant of all central division algebras over L of index n is (p1 p2 )n(n−1) .
(4)
2.3 Algebraic space–time codes from representations of orders Let now C = (E/L, σ, γ) be a cyclic division algebra of index n. We fix compatible embeddings of L and E into C, and identify L and E with their images under these embeddings. To construct matrices to serve as codewords, we consider C as an n-dimensional right vector space. The E-linear transformation of C given by left multiplication by an element c ∈ C results in an L-algebra homomorphism ρ : C → Mat(n, E), to which we refer to as the maximal representation. An element c = c0 + uc1 + · · · + un−1 cn−1 ∈ C can be identified via the maximal representation with the matrix c0 γσ(cn−1 ) γσ 2 (cn−2 ) . . . γ σ n−1 (c1 ) c1 σ(c0 ) γσ 2 (cn−1 ) . . . γσ n−1 (c2 ) (5) ρ(c) = . . .. .. .. .. .. . . . . cn−1 σ(cn−2 ) σ 2 (cn−3 ) · · · σ n−1 (c0 ) The determinant nr(c) = det(ρ(c)) and trace tr(c) = Tr(ρ(c)) define the reduced norm and reduced trace of c, respectively. Next, given a lattice Λ in C, we may use the maximal representation to define an injective map ρ Λ ,→ Mat(n, E) ⊂ Mat(n, C). Any finite subset X of ρ(Λ) or its transpose ρ(Λ)t will be a space–time lattice code. In the literature, cyclic-division-algebra based space–time codes are often referred to as algebraic space–time codes. Due to the division algebra structure the above matrices will be invertible by definition, and hence any algebraic space-time code constructed in this way will have full diversity. Example 1 Let E/L be a quadratic real extension of number fields, with L of class number 1, OL and OE = OL [ω] their respective rings of integers × and with the Galois group Gal(E/L) = hσi. We choose γ ∈ OL such that γ∈ / NmE/L (E × ), and define the cyclic division algebra C = (E/L, σ, γ) ∼ = E ⊕ uE,
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where u2 = γ. An algebraic space–time code constructed from the transpose of the maximal representation (5) is a finite subset x1 + x2 ω x 3 + x4 ω xi ∈ OL . X ⊂ γ(x3 + x4 σ(ω)) x1 + x2 σ(ω)
√ The choice of fields (E, L) = (Q(i, 5), Q(i)) and element γ = i gives rise to the Golden algebra and to the well-known Golden code [8]. The above setting, as well as Example 1, relate to the symmetric scenario, i.e., full lattices that can be efficiently decoded when nr = nt . The same codes can also be employed when nr > nt . There is no simple optimal decoding method for symmetric codes, however, when nr < nt 7 . Below, we will briefly introduce the block diagonal asymmetric space-time codes. The principles for our code constructions in the subsequent sections will follow those of the block diagonal codes, with the distinction of using natural orders instead of maximal ones. We refer the reader to [6] for more details. Let F ⊂ L ⊂ E be a tower of field extensions with extension degrees [E : L] = nr , [L : F ] = n, and [E : F ] = nt = nr n, and with Galois groups Gal(E/F ) = hτ i and Gal(E/L) = hσi = hτ n i. We fix a non-norm element nL r −1 × γ ∈ OL , and consider the cyclic division algebra C = (E/L, σ, γ) = ui E. i=0
Given any order O in C, we identify each element c ∈ O with its maximal representation ρ(c) and construct the following infinite block-diagonal lattice achieving the nonvanishing determinant property, provided that the base field F is either Q or quadratic imaginary [6]: ρ(c) 0 ··· 0 0 τ (ρ(c)) 0 ∈ Mat(nt , C) c ∈ O . L(O) = . . . .. .. .. n−1 0 ··· 0 τ (ρ(c))
Remark 2 The code rate [6] of a space–time code carved out from the infinite block-diagonal lattice above, that is, the ratio of the number of transmitted independent (complex) information symbols (=dimensions) to the number of channel uses, is – nn2r /nnr = nr , if the base field F is quadratic imaginary. – nn2r /2nnr = nr /2 , if the base field is F = Q. We point out that nr is the maximum code rate that allows for avoiding accumulation points at the receiving end with nr receive antennas; see the footnote below. 7 Having too few receive antennas will cause the lattice to collapse resulting in accumulation points, since the received signal now has dimension 2nr nt < 2n2t . Hence, partial brute-force decoding of high complexity has to be carried out.
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In summary, in order to construct an algebraic space–time code, we first choose a central simple algebra over a suitable base field and then look for a dense lattice in it. This amounts to selecting an adequate order in the algebra. As has already been mentioned, orders with small discriminants are optimal for the applications in space–time coding which makes maximal orders the obvious candidates. Unfortunately, they are in general very difficult to compute and may result in highly skewed lattices making the bit labeling a delicate problem on its own. Therefore, natural orders with a simpler structure have become a more frequent choice as they provide a good compromise between the two common extremes: using maximal orders to optimize coding gain, on the one hand, and restricting to orthogonal lattices to simplify bit labeling, encoding, and decoding, on the other hand.
3 Natural orders with minimal discriminant In this section we will consider the tower of extensions depicted in Figure 1. In
C = (E/L, σ, γ) =
nr L
ei E
i=1
nr
E nr
L n
√ F = Q( d) Fig. 1 Tower of Field Extensions.
order to get the nonvanishing determinant property, the base field F is chosen to be either the rationals or the imaginary quadratic extension Q(i). Let us now fix the base field and the extension degrees n and nr . Our goal will be to find an × explicit field extension E/L and a non-norm element γ ∈ OL such that E/F is a cyclic extension, (E/L, σ, γ) is a cyclic division algebra, and the norm of the discriminant disc(Onat /OF ) is the minimum possible among all cyclic division algebras satisfying the fixed conditions. Our findings are summarized in the following table and will be proved, row by row, in the subsequent theorems. Here α denotes a root of the polynomial X 3 − (1 + i)X 2 + 5iX − (1 + 4i).
10 F Q Q Q(i) Q(i) Q(i)
Amaro Barreal et al. n 1 2 2 2 3
nr 2 2 2 3 2
nt 2 4 4 6 6
rate 1 1 2 3 2
NmOF (disc(Onat /OF )) 32 56 56 318 · 1312 312 · 138
L Q √ Q( √ 5) Q(i, 5) √ Q(i, i 3) Q(i, α)
E √ Q(i 3) Q(ζ5 ) Q(i, ζ5 ) √ Q(i, i√3, α) Q(i, i 3, α)
γ -1 -1 i
√ 1+i 3 2
i(α − 1)
Table 1 Main results summarized.
√ Theorem 1 Let Q ⊂ E = Q( d), d ∈ Z square-free. Any cyclic division algebra (E/Q, √ σ, γ) satisfies NmZ (disc(Onat /Z)) ≥ 9, and equality is achieved for E = Q(i 3), γ = −1.
Proof The smallest possible quadratic discriminant over Q is 3, correspond√ ing to the field Q(i 3) = Q(ω), with ω a primitive cubic root of unity. Since all of the six units in √ Z[ω]× have norm 1, we conclude that −1 ∈ / NmZ[ω]/Z . Consequently, Q(i 3)/Q, σ, −1 is a division algebra and, using (3), NmZ (disc(Onat /Z)) = 9. t u
Theorem 2 Let Q ⊂ L ⊂ E with [E : Q] = 4 and [E : L] = 2. If (E/L, σ, γ) 6 is a cyclic equality for √ division algebra, then NmZ (disc(Onat /Z)) ≥ 5 , with L = Q( 5), E = Q(ζ5 ) and γ = −1, where ζ5 is a primitive 5th root of unity. √ Proof q The fields L and E can be uniquely expressed as L = Q( D), E = √ Q A(D + B D) , with A, B, C, D ∈ Z such that A is square-free and
odd, D = B 2 + C 2 is square-free, B, C > 0 and gcd(A, D) = 1 (see [9], [10]). We study the possible cases.
(i) If D ≡ 0 (mod 2), then disc(E/Z) = 28 · A2 · D3 . This expression takes its minimum value for D = 2, B = C = 1, and |A| = 1. Since NmL/Q (γ)2 ≥ 1 for all γ ∈ OL , using (3) we get NmZ (disc(Onat /Z)) ≥ 222 . (ii) If D ≡ B ≡ 1 (mod 2), then disc(E/Z) = 26 · A2 · D3 . The minimum value of the above expression is attained for D = 5 and |A| = B = 1, hence NmZ (disc(Onat /Z)) ≥ 212 · 56 . (iii) If D ≡ 1 (mod 2), B ≡ 0 (mod 2) and A + B ≡ 3 (mod 4), we have disc(E/Z) = 24 · A2 · D3 . Since 24 A2 D3 will take its minimum value for D = 5, B = 2, and A = 1, we have NmZ (disc(Onat /Z)) ≥ 28 · 56 . (iv) Finally, if D ≡ 1 (mod 2), B ≡ 0 (mod 2), A + B ≡ 1 (mod 4), A ≡ ±C (mod 4), we have disc(E/Z) = A2 · D3 . This expression attains its minimum for D = 5, B = 2, and A = −1, thus NmZ (disc(Onat /Z)) ≥ 56 .
The last case gives us the minimal natural order discriminant, corresponding to the fields E = Q(ζ5 ) and L = Q(ζ5 )+ = Q(ζ5 + ζ5−1 ), and to the element γ = −1. To conclude the proof, it suffices to show that −1 ∈ / NmE/L (E × ), + i.e., the algebra (Q(ζ5 )/Q(ζ5 ) , σ, −1) is division. Suppose that x ∈ Q[ζ5 ] with NmE/L (x) = −1. Then NmE/Q (x) = 1, thus x ∈ Z[ζ5 ]× and we can write x = ζv, where ζ is a root of unity and v ∈ L× (see [11, Prop. 6.7]). Consequently, −1 = NmE/L (x) = NmE/L (ζ) · NmE/L (v) = v 2 . But −1 is not a square in L. t u
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Theorem 3 Let Q(i) ⊂ L ⊂ E, with [E : Q(i)] = 4 and [E : L] = 2. Any cyclic division algebra (E/L, σ, γ) satisfies NmZ[i] (disc(Onat /Z[i])) ≥ 56 under √ these assumptions, and equality is achieved for L = Q(i, 5), E = Q(i, ζ5 ), and γ = i. √ √ √ Proof We observe that Q(i) ∩ Q( 5) = Q and Q(i, 5) ∩ Q(ζ5 ) = Q( 5). Thus, if we prove that i ∈ / NE/L , the result will be a consequence of the computations in Theorem 2. By the Hasse Norm Theorem (see [7, Thm. 32.9]), it suffices to show that i ∈ / NmEt /Lq (Et ) for some primes q of L and t of E extending q. If p5 is one of the two primes with norm 5 in Z[i], it is not difficult to verify that p5 OL = q25 , q5 OE = t25 , NmL (q5 ) = NmE (t5 ) = 5 and disc(E/L) = q5 , so the only prime that ramifies in the extension E/L is q5 . Hence, the obvious choice is Et5 /Lq5 , a totally and tamely ramified cyclic local extension of degree 2, with |OLq5 /q5 OLq5 | = |OL /q5 OL | = 5. × In order to see that i ∈ / NmEt5 /Lq5 (OL ), we determine NmEt5 /Lq5 (Et×5 ). q 5
Let π ∈ OEt5 be a uniformizer; then, Et×5 is the group generated by π and × × OE(t and NmEt5 /Lq5 (Et×5 ) = hNmEt5 /Lq5 (π)i NmEt5 /Lq5 (OE ). The group t 5) 5
× × NmEt5 /Lq5 (OE ) is a subgroup of OL and, by Local Class Field Theory (see t q 5
5
× [12, Thm. 1.1]), we have L× q5 /(NmEt5 /Lq5 (Et5 )) ' Gal(Et5 /Lq5 ) = C2 , as well as i h h i × × × ) = e (Et5 |Lq5 ) = Et×5 : L× OL : NmEt5 /Lq5 (OE (6) q5 OEt5 = 2. t5 q5 n o (1) (1) × x ≡ 1 (mod q ) , Let UE = x ∈ O5× x ≡ 1 (mod t5 ) , UL = x ∈ OL 5 q 5
(1)
× ζ4 ∈ Et5 be a 4th root of unity and write OE = hζ4 iUE . Since the ext5 tension is totally ramified, the residue fields of Et5 and Lq5 agree and, so, (1) (1) NmEt5 /Lq5 (ζ4 ) = ζ42 = −1. On the other hand, NmEt5 /Lq5 (UE ) = UL , as × × otherwise the group OL / NmEt5 /Lq5 (OL ) would contain an element x ˜ with q t 5
5
ord(˜ x) = 5k some k, contradicting (6). We conclude that NmEt5 /Lq5 (Et×5 ) = (1)
(1)
× hNmEt5 /Lq5 (π), −1iUL and, more specifically, NmEt5 /Lq5 (OE ) = h−1iUL , t 5
(1)
with ζ4 ∈ / h−1iUL . Since γ = i has multiplicative order 4 in the residue field, we deduce that γ ∈ / NmE/L (E × ). t u Theorem 4 Let F = Q(i) ⊂ L ⊂ E with [E : Q(i)] = 6 and [E : L] = 3. Any cyclic division algebra (E/L, σ, γ) satisfies NmZ[i] (disc(Onat /Z[i])) ≥ 318 ·1312 . √ The lower bound is achieved for L = Q(i, i 3), E = L(α),√ with α a root of f (X) = X 3 − (1 + i)X 2 + 5iX − (1 + 4i), and γ = ω = 1+i2 3 . Proof Equation (3) in Lemma 2 tells us that it suffices to find explicit E, L and γ in the given conditions, such that the discriminant of E/Z[i] is smallest × possible and γ k ∈ OL \ NmE/L for k = 1, 2. Step 1: We start by finding the smallest possible discriminant over Z[i] for cyclic extensions of degree 6. For each rational prime p, we write pZ[i] = p2p ,
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pp p0p or pp = (p) for the decomposition of p (ramified, split or inert, respectively) in Z[i], and we denote by L2 , L3 and E6 = L2 L3 cyclic extensions of degree 2 and 3 over Q(i), and their compositum, respectively. Class Field Theory will provide us with fields L2 and L3 of smallest discriminants, and we will use them to compute the discriminant of E6 . Since F = Q(i), it is enough to search for candidates among of ray class fields modulo Q nsubfields ideals m of Z[i]. For an ideal m = p p ⊂ Z[i], let Um be the group of principal fractional ideals in Q(i) generated by elements a ∈ Q(i) such that vp (a − 1) ≥ np for all primes p dividing m, and Im(Um ) its image in (Z[i]/m)× . We know that for each ideal m there exists a unique extension L of Q(i) such that Gal(L/Q(i)) ' (Z[i]/m)× /Im(Um ), and the primes that ramify in the extension L/Q(i) all divide m. L3 : In order to obtain a cubic extension we need 3 to be a factor of |(Z[i]/m)× |, leaving the choices m = π or m = π · 32 , where π is a product of prime ideals with norm congruent to 1 modulo 3. The smallest possible norm for primes in m is 13 = NmZ[i] (p13 ) = NmZ[i] (p013 ). Taking m = p13 yields |(Z[i]/m)× /Im(Um )| = 3 and, consequently, there exists a cubic extension L3 of Q(i) that ramifies exactly at p13 . Furthermore, since the ramification index is 3 and hence relatively prime to p13 , by a theorem of Dedekind the discriminant of the extension is disc(L3 /Q(i)) = p213 , and NmZ[i] (L3 /Q(i)) = 132 . This corresponds to the smallest possible cubic discriminant over Q(i) (we observe that the last calculation also proves that p13 is not the discriminant of a quadratic extension over Q(i)). The corresponding sextic field over Q has discriminant 26 · 132 , and by [13, Table 1], the extension L3 corresponds to the polynomial f (X) = X 3 − (1 + i)X 2 + 5iX + (−1 − 4i), say L3 = Q(i, α), where α is a root of f , with discriminant (2 − 3i)2 . Consequently, p13 = (2 − 3i) is the only prime ramifying in L3 /Q(i), and p13 OL3 = q313 , with NmL3 (q13 ) = 13. L2 : For a quadratic extension, we need to consider the cases (m, p13 ) = 1 and (m, p13 ) 6= 1 separately. In the first case, the best possible choice is m = p3 = (3), resulting in |(Z[i]/m)× /Im(Um )| = 2. Thus, p3 is the discriminant of a quadratic extension of Q(i); indeed, it is the discriminant of L2 = √ Q(i, i 3) = Q(ζ12 ), where ζ12 is a 12th root of unity. If m = p13 p for some ideal p 6= (1), the best we can do is to take p = p22 . Then Um = {±1, ±i} ∪ × { u0 + (2 − 3i)(1 + i)r | u0 ∈ {±1, ±i} , r = √ 2, 3}, and |(Z[i]/m) /Im(Um )| = 0 2. The corresponding field is L2 = Q(i, 3 + 2i), whose discriminant ideal is p13 · p22 . Taking E6 = L2 L3 , we get disc(E6 /Q(i)) = p33 p413 , whose norm equals NmZ[i] (disc(E6 /Q(i))) = 36 · 134 . For E60 = L02 L3 , we get disc(E60 /Q(i)) = p513 p62 , with NmZ[i] (disc(E60 /Q(i))) = 135 · 26 . Note that the factor p513 comes from the fact that E60 /L3 is tamely ramified, and hence its discriminant is the ideal s22 s13 , where s22 = p2 OL3 and s13 is the prime ideal above 13 in L3 . Since 135 · 26 > 36 · 134 , we conclude that the smallest possible discriminant over Z[i] for cyclic extensions of degree 6 is 36 · 134 , achieved in the extension E6 = L2 L3 .
Natural orders for asymmetric space–time coding: minimizing the discriminant
13
h √ i The involved rings of integers are O2 = Z i, i+2 3 and O3 = Z[i, α], where √ IrrQ(i) i+2 3 = X 2 − iX − 1 and α is as above. To compute O3 , we observe that disc(f ) = (2 − 3i)2 = disc(1, α, α2 ) = disc(O3 /Z[i]) · [O3 : Z[i, α]]2 . Since the extension L3 /Q(i) ramifies at p13 , we conclude that O3 = Z[i, α].
Step 2: Next, we search for non-norm element γ of smallest possible norm. Equation (4) in Lemma 3 provides us with a lower bound for NmO2 (γ). In order to obtain it, we need to find a pair of smallest primes in E6 /L2 . Computing the factorization of primes8 and relative discriminants of the extensions involved, we find the following pairs of smallest primes. √ 3) and q3 = O2 : A pair of smallest primes in O2 are q2 = p2 , (1+i)(1+ 2 √ −3i+ 3 3, , of respective norms 4 and 9. 2 O3 : Since the primes above p7 and p11 have norms at least 49 and 121, respectively, a pair of smallest primes in O3 are s2 = p2 O2 and s13 = √ (1−i)(1+ 3+2i) 2 − 3i, − 2 , of respective norms 23 and 13. 2 The discriminants of the extensions involved are summarized in Table 2 below.
E6 L3 L2
disc(·/Q(i)) p33 p413 p213 p3
NmZ[i] 36 · 134 132 32
disc(·/L2 ) q213 q02 13
NmO2 134
disc(·/L3 ) s3
NmO3 36
Table 2 Relative discriminants of the field extensions involved.
If we let hσi = Gal(E6 /L2 ), then equations (3) and (4) and the above computations, tell us that any element γ ∈ O2 satisfying equation (2) –so that (E/L2 , σ, γ) is a division algebra with natural order Onat –, will satisfy the following inequality: NmO2 (disc(Onat /O2 )) = NmO2 ((q13 q013 )6 · γ 6 ) ≥ NmO2 (q2 q3 )6 = 46 · 96 . Since 1312 > 46 · 96 , there are no restrictions, and our searched for non-norm element γ could be a unit. Consequently, NmZ[i] (disc(Onat /Z[i]) willl achieve its smallest possible values if we can find a unit γ ∈ O2× such that γ k ∈ / NmE6 /L2 , k = 1, 2, in which case we will have NmZ[i] (disc(Onat /Z[i])) = 318 · 1312 , and the theorem will be proved. Step 3: To simplify notation, we set E6 = E, L2 = L, and O2 = OL . We will use Hasse Norm theorem, and a local argument analogous to the one used in √ × Theorem 3, to prove that the unit γ = 1+i2 3 ∈ OL satisfies γ k ∈ / NmE/L , k = 8
For the complete computations see A.
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1, 2. As disc(E/L) = q213 , the only prime that ramifies in the extension E/L is q13 . If t13 is a prime of E extending q13 , the extension Et13 /Lq13 is a totally and tamely ramified cyclic extension of degree 3, with |OLq13 /q13 OLq13 | = |OL /q13 OL | = 13. Let π ∈ OEt13 be a uniformizer, Then, × NmEt13 /Lq13 (Et×13 ) = hNmEt13 /Lq13 (π)i NmEt13 /Lq13 (OE ), t 13
× L× q13 /(NmEt13 /Lq13 (Et13 )) ' Gal(Et13 /Lq13 ) = C3 ,
and h × OL q
13
Let
i h i × × × × : NmEt13 /Lq13 (OE ) = e (E |L ) = E : L O = 3. (7) t13 q13 q13 Et t13 t 13
(1)
o × x ≡ 1 (mod t ) , x ∈ OE 13 t13 o n × = x ∈ OL x ≡ 1 (mod q13 ) , q
UE = (1)
UL
13
n
13
(1) hζ12 iUE ,
× and write OE = where ζ12 ∈ Et13 is a 12th root of unity, The t13 extension being totally ramified, the residue fields of Et13 and Lq13 agree (1) (1) 3 . On the other hand, NmEt13 /Lq13 (UE ) = UL ; and NmEt13 /Lq13 (ζ12 ) = ζ12 × × else, the group OL / NmEt13 /Lq13 (OL ) would contain an element x ˜ with q t 13
13
ord(˜ x) = 13k , some k, contradicting (6). Thus, we have NmEt13 /Lq13 (Et×13 ) = (1)
(1)
× 3 3 hNmEt13 /Lq13 (π), ζ12 iUL , and NmEt13 /Lq13 (OE ) = hζ12 iUL , with ζ12 ∈ / t (1)
√
13
3 hζ12 iUL . Since γ = 1+i2 3 has multiplicative order 6 in the residue field, we can conclude that neither γ nor γ 2 are in NmE6 /L2 (E6× ). t u
Theorem 5 Let F = Q(i) ⊂ L ⊂ E with [E : Q(i)] = 6 and [E : L] = 2, and let (E/L, σ, γ) be a cyclic division algebra. Then NmZ[i] (disc(Onat /Z[i])) ≥ 312 · 138 with equality for L = Q(i, α) with α a root of the polynomial√f (X) = X 3 − (1 + i)X 2 + 5iX − (1 + 4i), E = LL2 , where L2 = Q(i, i 3), and γ = i(α − 1). Proof As we saw in the proof of Theorem 4, Step 1, the choice of E ensures the minimality of its discriminant over Z[i], among all possible discriminants of cyclic sextic extensions over Q(i). Hence, it suffices to prove that i(α − 1) × is a non-norm element in OL . Let t3 be a prime of E extending s3 , the only prime that ramifies in the extension E/L. If, using Hasse Norm Theorem, we × × can show that i(α − 1) ∈ OL \ NmEt3 /Ls (OE ), we will be done. s t 3
3
3
(1)
× Let ζ12 ∈ Et13 be a 12th root of unity and UL = {x ∈ OL |x ≡ s3 1 (mod s3 )}. Using a local argument analogous to the ones used in theorems (1) × 2 3 and 4, we get NmEt3 /Ls3 (OE iUL . We are left with showing that ) = hζ12 t3 i(α − 1) is a unit in OL such that its image in the residue field is not in (1) 2 hζ12 iUL . It is easy to see that if q ∈ Q(i), then NmL/Q(i) (α − q) = f (q). Thus, since f (1) = −1, ξ = α − 1 is a unit. Also, ξ cannot be a root of unity;
Natural orders for asymmetric space–time coding: minimizing the discriminant
15
otherwise L would be abelian over Q, which it is not, as the primes over 13 split in different ways. Further, the image of ξ in the residue field is 1, which (1) 2 implies that the image of the unit iξ is 5 ∈ / hζ12 iUL , since 5 is not a square modulo 13. Thus, the choice γ = i(α − 1) gives us the required result. t u 4 Conclusions In this article we have introduced the reader to a technique used in multipleinput multiple-output wireless communications known as space–time coding. Within this framework, we have shown how to construct well-performing codes from representations of orders in central simple algebras, explaining why it is crucial to choose orders with small discriminants. While maximal orders achieve the minimum discriminant, we have motivated why in practice it may sometimes be favorable to use the so-called natural orders instead. For the base fields F = Q or F imaginary quadratic (corresponding to the most typical signaling alphabets), and pairs of extension degrees (nt , nr ) in an asymmetric channel setup, we have computed an explicit number field × extension (E/L) and an element γ ∈ OL giving rise to a cyclic division algebra whose ideal norm of the corresponding natural order Onat , viewed as an OF module, achieves the minimum possible among all cyclic division algebras with the same degree assumptions. This way we have produced explicit space–time codes attaining the optimal coding gain among codes arising from natural orders. Acknowledgements A. Barreal and C. Hollanti are financially supported by the Academy of Finland grants #276031, #282938, and #283262, as well as a grant from the Finnish Foundation for Technology Promotion. The authors thank Jean Martinet for his useful suggestions.
References 1. V. Tarokh, N. Seshadri, A. R. Calderbank, Space–Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction, IEEE Transactions on Information Theory 44 (2) (1998) pp. 744–765. 2. J.-C. Belfiore, G. Rekaya, Quaternionic Lattices for Space–Time Coding, Proceedings of the IEEE Information Theory Workshop, Paris (2003). 3. B. A. Sethuraman, B. S. Rajan, V. Shashidhar, Full-Diversity, High-Rate Space–Time Block Codes from Division Algebras, IEEE Transactions on Information Theory 49 (10) (2003) pp. 2596–2616. 4. C. Hollanti, J. Lahtonen, H.-f. Lu, Maximal Orders in the Design of Dense Space-Time Lattice Codes, IEEE Transactions on Information Theory 54 (10) (2008) pp. 4493– 4510. 5. R. Vehkalahti, C. Hollanti, J. Lahtonen, K. Ranto, On the Densest MIMO Lattices From Cyclic Division Algebras, IEEE Transactions on Information Theory 55 (8) (2009) pp. 3751–3780. 6. C. Hollanti, H.-f. Lu, Construction Methods for Asymmetric and Multiblock Space– Time Codes, IEEE Transactions on Information Theory 55 (3) (2009) pp. 1086–1103. 7. I. Reiner, Maximal Orders, London Mathematical Society Monographs New Series 28 (2003).
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8. F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo, Perfect Space–Time Block Codes, IEEE Transactions on Information Theory 52 (9) (2006) pp. 3885–3902. 9. K. Hardy, R.H. Hudson, D. Richman, K. S. Williams, N. M. Holz, Calculation of Class Numbers of Imaginary Cyclic Quartic Fields, Carleton-Ottawa Mathematical Lecture Notes Series 7 (1986). 10. R. H. Hudson, K. S. Williams, The Integers of a Cyclic Quartic Field, Rocky Mountains Journal of Mathematics 20 (1) (1990) pp. 145–150. 11. J. Milne, Algebraic Number Theory, Graduate course notes v2.0 (2014) http://www.jmilne.org/math/coursenotes/ . 12. J. Milne, Class Field Theory, Graduate course notes v4.02 (2013) http://www.jmilne.org/math/coursenotes/ . 13. A.-M. Berg´ e, J. Martinet, M. Olivier. The Computation of Sextic Fields with a Quadratic Subfield, Mathematics of Computation 54 190 (1990) pp. 869–884.
A Factorization of Primes We compute the factorization of primes and relative discriminants in the extensions involved in Theorem 4. The notation is explained below the table.
O6 O3
p2 O3 = s2
(3)O3 = s3
O2
p2 O2 = q2
(3)O2 = q23
Z[i] Z
(1 + i)2 = p22 2
(3) = p3 3
p5 O3 = s5 p05 O3 = s05 p5 O2 = q5 p05 O2 = q05 (2 ± i) = p5 p05 5
(7)O2 = q7
(11)O2 = q11 q011
(7) = p7 7
(11) = p11 11
q13 O6 = t313 p13 O3 = s313 p013 O3 = s013 p13 O2 = q13 q013 000 p013 O2 = q00 13 q13 (2 ± 3i) = p13 p013 13
Table 3 Factorization of prime ideals.
q2 = p2 , q3 = 3,
√ (1+i)(1+ 3) , 2
NmO2 (·) = 4;
√ (−3i+ 3) , 2
NmO2 (·) = 9.
√ √ 2 − i i+2 3 − 1 , q5 = 2 + i, i+2 3 q7 =
7,
√ 2 i+ 3 2
−i
√ i+ 3 2
√ q11 = 11, i+2 3 − 6i − 3 ,
s13
= 2 − 3i,
q013 = 2 − q000 13 = 2 +
−2 ,
NmO2 (·) = 25. NmO2 (·) = 74 .
√ q011 = 11, i+2 3 − 6i + 3 ,
q00 13
√ (1−i)(1+ 3+2i) 2
√ √ 2 − i i+2 3 − 1 2 − i, i+2 3
−1 ,
√ = 2 − 3i, i+2 3 − 7i − 2 , √ = 2 + 3i, i+2 3 − 7i − 2 ,
q13
q05 =
√ 3i, i+2 3 √ 3i, i+2 3
− 7i + 2 , − 7i + 7 ,
NmO2 (·) = 121.
NmO2 (·) = 13. NmO3 (·) = 13.
