2006 IEEE Information Theory Workshop, Punta del Este, Uruguay, March 13-17, 2006
A New Tool: Constructing STBCs from Maximal Orders in Central Simple Algebras Jyrki Lahtonen
Camilla Hollanti
TUCS Lab. of Discrete Mathematics University of Turku FIN-20014 Turku, Finland Email:
[email protected]
Department of Mathematics University of Turku FIN-20014 Turku, Finland Email:
[email protected]
Abstract-A means to construct dense, full-diversity STBCs from maximal orders in central simple algebras is introduced for the first time. As an example we construct an efficient ST lattice code with non-vanishing determinant for 4 transmit antenna MISO application. Also a general algorithm for testing the maximality of a given order is presented. By using a maximal order instead of just the ring of algebraic integers, the size of the code increases without losses in the minimum determinant. The usage of a proper ideal of a maximal order further improves the code, as the minimum determinant increases. Simulations in a quasi-static Rayleigh fading channel show that our lattice outperforms the DAST-lattice due to the properties described above.
I. INTRODUCTION We are interested in the coherent multiple input-single output (MISO) case where the receiver perfectly knows the channel coefficients. The received signal is hlxkXkxn ± nlixnl,
Ylxn
where X is the transmitted codeword taken from the SpaceTime Block Code (STBC) C, h is the Rayleigh fading channel response and the components of the noise vector n are i.i.d. complex Gaussian random variables. A lattice is a discrete finitely generated free Abelian subgroup L of a real or complex finite dimensional vector space V, called the ambient space. In the space-time setting a natural ambient space is the space M n (C) of complex n x n-matrices. The receiver, however, (recall that we work in the MISO setting) sees vector lattices instead of matrix lattices. When the channel state is h, the receiver expects to see the lattice hL. From the pairwise error probability (PEP) point of view, it is well-known that the performance of a space-time code is dependent on two parameters: diversity gain and coding gain. Diversity gain is the minimum of the rank of the difference matrix X-X' taken over all distinct code matrices X, X' c C, also called the rank of the code C. When C is full-rank, the coding gain is proportional to the determinant of the matrix (X-X') (X-XI)H, where H denotes the Hermitian transpose. The minimum of this determinant taken over all distinct code matrices is called the minimum determinalnt of the code C. When a code is a subset of a lattice L in the ambient space Mn(C), the ralnk criterion states that any non-zero matrix 1-4244-0036-8/06/$20.00
©)2006 IEEE
in L must be invertible. This follows from the fact that the difference of any two matrices from L is again in L. It is widely known how the so called Alamouti design represents multiplication In the ring of quaternons. As the quaternions form a division algebra, such matrices must be invertible, i.e. the resulting STBC meets the rank criterion. Matrix representations of other division algebras have been proposedas STBC codes at least in Ill [2],131 141 and (though
codes at in []2 e]] and (though proposedpascStBy without explicitly saying so)les. [5]. The most recent work (13]4]
and [5]) has concentrated on adding multiplexing gain (i.e.
MIMO applications) and/or combining it with a good minimum determinant. We do not seek any multiplexing gains, but want to improve upon e.g. the DAST-lattices introduced in [2] by using not only non-commutative division algebras, but the maximal orders within them. The usage of division algebras has been of the utmost interest in the recent study, as they naturally produce families of linear, full-rank codes. By choosing the elements in the code matrices from a maximal order instead of just picking them from the ring of algebraic integers - albeit in some cases these may collapse - the size of the code can be increased without losses in the minimum determinant. By further requiring the elements to belong to a proper ideal of a maximal order the minimum determinant increases and hence, after scaling, denser lattices are produced.
II. LATTICE CONSTRUCTION
The set {a-+ a2i + a3j + a4kJ ai C R Vi}, where i2 = j2 = = _1, ij = k, is recalled as the ring of Hamiltonian quaternions. We shall use extension rings of the Gaussian integers g = {a + biIa, b C Z} inside a given division algebra as they nicely fit with the popular 16-QAM and QPSK alphabets. Let ( = C-iI4 = (1 + i)/x/2 be a primitive 8th root of unity. Our main example is the division algebra H = Q(,) G Q(,)j. As zj = jz* for all complex numbers z, and as the field Q(Q) is stable under the usual complex conjugation (*), the set H is a subskewfield of the quaternions.
As always, multiplication from the left by a non-zero element of the division algebra A is an invertible Q(i)-linear mapping with Q(i) acting from the right. Therefore its matrix with respect to a chosen Q(i)-basis 13 of A is also invertible. The division algebra H has the set liH ={1, t, , jt} as a
322
natural Q(i)-basis. Thus we immediately arrive (see also [1]) at the following matrix representation of the division algebra H. Proposition 2.]: Let the variables C1, C2, C3, C4 range over all the elements of Q(i). The division algebra H can be identified via an isomorphism X with the following ring of matrices H ( Cl iC2 4 3 l
t M = (cl,c2 C3rC4 = 1 CC3 t c4
C1 tc4 C3
i4 C1 -32
-tC2
J|
'1 ) In particular the determinants of these matrices are non-zero whenever at least one of the coefficients C1, C2, C3, C4 is non* zero. In order to get STBC-lattices and useful bounds for the minimum determinant we need to identify suitable subrings R of the algebra H. We shall do this by placing certain restrictions for the elements C , c2, c3, C4. Later on, in section III, we shall show that one of these restrictions produces a maximal order. The g-module spanned by our earlier basis P3H is a ring C of the required type. We call this ring the ring of Lipschitz' integers of H. The ring 4(z) consists of those matrices of H that have all the coefficients cl, c2, c3, C4 C 5. However, C is not maximal among the rings satisfying our requirements. The ring of Hurwitz' integral quaternions also has an extension of the prescribed type inside H. This ring, denoted by A, is the right 5-module generated by the basis L3H,ur = f P, pj,j, where p = (1 + i + j + k)/2. For future use we express the ring N in terms of the basis li3H of Proposition 2.1. It is not difficult to show that the quaternion q = C1 A + C2 + jC3 +Aj(C4 is an element of N, if and only if the coefficients ct, t = 1, 2, 3, 4 satisfy the requirements (1 + i)ct C 5 for all t and Cl + C3, C2 + C4 C 5. As the ideal generated by 1 + i is of index two in 5, we see thatf is an additive subgroup of index four in N. We summarize these findings in Proposition 2.2. The bound on the minimum determinant is a consequence of the fact that all the elements of g have norm at least 1. rin sofmatricesf followin rings Pro.osition.2: The following 22 The of matrices form STBC-lattices of minimum determinant 1. LL = {M(cl, C2, C3, C4)ICl, C2, C3, C4 C 5}, 1 +i Lh = {M(c1,c2,C3,C4)jC1,C2,C3,C4 C 2 * Cl + C3 C g, C2 + C4 C g}Remark 2.1: The lattice LL is a more developed case from the so-called quasi-orthogonal STBC suggested e.g. in [6]. The matrix of LL can be found as an example also in [3], but no optimization has been done there by using, for example, maxima order as weshall o here A drawback of the lattice L.c is that in the ambient space of the transmitter it is isometric to the rectangular lattice z 8. The rectangular shape does carry the advantage that the sets of information carrying coefficients of the basic matrices are
simple and all identical (this is useful in e.g. sphere decoding), but this shape is very wasteful in terms of transmission power. Geometrically denser sublattices of Z8, e.g. the diamond lattice E8 are well known (cf. e.g. [7]). However, we must be careful when picking the copies of the sublattices, as it is the minimum determinant we want to keep an eye on. The units of the ring LL are exactly the non-zero matrices, whose determinants have the minimal absolute value one. Thus an intuitive way to find a sublattice with a better minimum where I CR is aproper is totake the lattice 0(l), determinant and [4]. Even earlier, ideals This idea has appeared in [1] ideal. of rings of algebraic integers were used in [8] to produce dense lattices. The diamond lattice E8 can be described in terms of the Gaussian integers as follows (cf. [9]): 1 E{(C, c2, 3 c4) 4 + f = Ct + ,
t=l
By our identification of quadruples (c 1, C2, C3, C4) C 54 and elements of H it is readily verified that A = (1 + i)E8 has { 2, (1 + i) + (1 + i)t, (1 + i)t + (1 + i)j, 1 + ( +Aj + j(5} C as a g-basis, whence the set {1 + i, 1 + (, ( + j, p + pt} C n is a g-basis for E8. By another simple computation we see +), i.e. E8 is the left ideal of the ring 'H that E8 = -(1 A generated by 1 + (. Proposition 2.3: The lattice
j2C
LE8
{M(c1,C2,C3,C4) C L| C1 A-I t
2,3,4,
t=1
Ct A-,
Ct c 25}
is an index 16 sublattice of LL. Furthermore, the minimum determinant of LE8 is 64. Proof: Let Ml = M(1, 1,0,0) be the matrix q$(1 + () under the isomorphism of Proposition 2.1. We see that det(MIM,*) = 4. By the preceding discussion any matrix A of the lattice LE8 is of the form A = MMI(1 + i), where M is a matrix from LA. Thus, det AA* = 16 det(MiM) det(MM*) and the claim on the minimum determinant follows from Proposition 2.2. We see that the coefficient cl can be chosen arbitrarily within g. The coefficients C2 and C3 then must belong to the coset cl + I, and C4 must be chosen such that c1 + C2 + C3 + C4 E 2 = 12. As I is of index two in 5, we see that the index of LE8 in LL is 16 as claimed. U ORDERS III. CYCLIC ALGEBRAS AND The theory of cyclic algebras and their representations as matrices are thoroughly considered in [3]. We are only going to recapitulate the essential facts here. In the following, we consider number field extensions F/F, where F denotes the base field. F* (resp. E*) denotes the set of non-zero elements of F (resp. F). Let F/F be a cyclic field extension of degree n with Galois group Gal (F/F)=
323
Let us illustrate the above definition by the following (o-f, where a is the generator of the cyclic group. Let A = (E/F, o-, -y) be the corresponding cyclic algebra of degree n, example. that is Example 3.1: (a) Orders always exist: If M is a full RA = E (D uE (D u2E (D . .. D Un- AIE,F Fea2 . 1,lattice in A, i.e. FM = A, then the left order of M defined with u C A such that xa = auo(x) for all x C E and n= as 0i(M) = {x c AlxM C M} is an R-order in A. The -' c F*. An element a = xo + ux1 + ... + un-1 Xn-1 C A right order is defined in an obvious way. (b) If A Mn(F), the algebra of all n x n matrices over has the following representation as a matrix A = F, then A =Mn(R) is an R-order in A. n-l (x) x OJ(Xn-i) -OJ2(xn-2) .. (c) Let a E A be integral over R, that is, a is a zero of a -7n-l (X) Xl (xn-l) -Y(72(xn-1) monic polynomial over R. Then the ring R[a] is an R-order X1 o-5(X0)
uT(X1)
X2
X
(xn-1 0(n-2)
u(xJn (X0) uJ (X3)
J2(Xn-3)
'yuJ (X3) 1 .
...
. CJn-l(x0)
Let us compute the third column as an example:
a22-4au2 =
x0a2 ±axia2 + ...±Un-1 Xni2 uo(X0)a A- 2u(X1)a - +- 07(Xn1)U
(Xn-l
2 202 U_~_2 (X0 (Xj A*.. A- z 2 (xn1), = 2o2f (X0) A- a2 (JxZ1) and hence for the third column we get Tthe vector
a_u2(Xn-2), a.OJ2(Xn-1),
2(X0),...
(Xn3))T.
Definition 3.1: An algebra A is called simple, if it has no non-trivial ideals. An F-algebra A is central if its centre Z(A) = {a c Alaa'= a'a Va' Al = F. Definition 3.2: The determinant (resp. trace) of the matrix A is called the reduced norm (resp. reduced trace) of an element a E A and is denoted by nr(a) (resp. tr(a)). t Remark 3.]: The connection with the usual norm map NA/F(a) (resp. trace map TA/F(a)) and the reduced norm (resp. reduced trace tr(a)) of an element a E A is NA/F(a) = (nr(a))n (resp. TA/F(a) = ntr(a)), where n is the degree of E/F. In the preceding section we have attested that the algebra H is a division algebra. The next proposition provides us with a sufficient condition when an algebra is indeed a division
elementra ATand isndenoted byitr(a) (reusp.lor( 31
nr,a)
algebra. Proposition 3.1: The algebra A = (E/F, a, i) of degree n is a division algebra if the smallest factor t C Z of n such that jt is the norm of some element in E* is n. * We are now ready to present some of the basic definitions and results from the theory of maximal orders. The general theory of maximal orders can be found in [10]. Let R denote a Noetherian integral domain with a quotient field F, and let A be a finite dimensional F-algebra. Definition 3.3: An R-order in the F-algebra A is a subring A of A, having the same unity element as A, and such that A is a finitely generated module over R and generates A as a linear space over F. As usual, a A-order in A is said to be maximal, if it is not properly_contained in any other A-order in A. If the integral closure R of R in A4 happens to be an R-order in A4, then ft is automatically the unique maximal ft-order in A.
in the F-algebra F[a]. (d) Let R be a Dedekind domain, and let E be a finite separable extension of F. Denote by R the integral closure of R in E. Then R is an R-order in E. In particular, taking R = Z, we see that the ring of algebraic integers of E is a Z-order in E. Proposition 3.2: Let F be an algebraic number field, A a finite dimensional semisimple algebra over F and A be a Zorder in A. Let OF stand for the ring of algebraic integers of F. Then F OFA is a OF-order containing A. As a consequence, a maximal Z-order in A is a maximal OF-order as well. U In Section IV some facts from the local theory of orders are required. Let us first define the ring ZP. Definition 3.4: For a rational prime p let Z p denote the ring
Qlr, s c Z, gcd(p, s) = 1}. s ZP is a discrete valuation ring with the unique maximal ideal pz. If A0 is a Z-order we use the notation A = Z A. ZP =
r E
Let S denote an arbitrary ring with identity. The Jacobson radical of S is the set Rad(S) = {x c SlxM = (0) for all simple left S-modules M}. Definition 3.6: Let m = dimFA. The discriminant of the R-order A is the ideal d(A) in R generated by the set C Am}.
Definition 3.5:
{det(tr(xixj))iW =,(xl, ...,x-m) It is clear that if A C F then d(F)Id(A) and, moreover, A = F if and only if d(F)
d(A).
Proposition 3.3: Let {a1, ...,,a} be
A and IF dd = dec Let be order A. Then any containing =j. det(tr(ajaj))T C dF Rfal,...,am} C A. A The following proposition gives us a useful tool to find the maximal orders within a given algebra. Proposition 3.4: Let A be an R-order in A. For each a C A we have nr(a) C R, tr(a) C R. C Proposition 3.5: Let F be a subring of A containing R such that FF = A, and suppose that each a F is integral over f. Then F is an f-order in A. Conversely, every R-order in U A4 has these properties. A ft-order 3.6: Every in is contained in a maxiCorollalry mal R-order in A4. There exists at least one maximal R-order in A. v
324
an F-basis of
Proposition 3.7: Let A be a simple algebra over F and M a finitely generated OF-module such that FM = A. Then there exists an element s c OF \ {0} such that s * 1 C M. Moreover, 01(M) = {b c s-1MJ bM < M} < s-1M. * Proposition 3.8: The prime ideals P of A and the prime ideals P of R are in one-to-one correspondence, given by P = R U X, 2 C PA. exactly the products products of primeideals. ideals of A are exactly prime ndeals. (i)(i) The ideals
must be an element of Z[V'2]. We may thus conclude that all the coefficients m1,f, £ = 0, 1, 2,3 are integers or halfintegers, and that the pairs m1,0,m1,2 (resp. m1,1,m1,3) must be of the same type, i.e. either both are integers or both are half-integers. A similar study of tr(qj) and tr(qjt) shows that the same conclusions also hold for the coefficients M2,et = 0, 1, 2, 3. Because Z[(] C A, replacing q with any quaternion of the form q-w, where w C Z[(] will not change the resulting order . Thus we may assume that the coefficients =
number such tmph Th The numballwsrs numbers mp are number mp replchnthat qwtha PAfo=sm 'PP mp..... mpar divisors of n and, except for finitely many prime ideals P of R, MP 1. (iii) d(A) is independent of the choice of the maximal order A. Moreover, mp > 1 implies that Pld(A). Proposition 3.9: Let P be a prime ideal of R and r be an R-order such that Fp is not a maximal Rp-order. Then there exists an ideal I > PF of F for which O1(I) > F. * Remark 3.2: The algebra H can also be viewed as a cyclic division algebra. As it is a subring of the Hamiltonian quaternions, its center consists of the intersection H n R = Q(V\2). Also Q(Q) is an example of a splitting field of H. In the notation above we have an obvious isomorphism
0, 1,2,3 all belong to the set {0, 1/2}. Similarly, if need be, - w'j for some w' c Z[~] allows us to assume that the coefficients £ = 0, 1, 2, 3 also all belong to the set {0, 1/2}. Further replacements of q by q-p or q-p~ then permit us to restrict ourselves to the case m2,f = 0, for all £ = 0, 1, 2, 3. If we are to get a proper extension of A, we are left with the cases q = (1 + i)/2, q = ,(1 + i)/2 and q = (1 + ,)(1 + i)/2. We immediately see that none of these have reduced norms in Z[v/2], so we have arrived at a contradiction. U IV GENERAL ALGORITHM FOR TESTING THE
=
H
(Q() /Q(V2-), o-, -1),
where a is the usual complex conjugation. Next we prove that the lattice LE8 =7-(1 A-+ ) is optimal within the cyclic division algebra H in the sense that it corresponds to a proper ideal of a maximal order in H. Proposition 3.10: The ring
7-= { q = c1 + (C2 + [C3 + ji(C4 C*.... , C4 C Q(i),
(1 +Ai)Ct C Z[i] Vt,cl + c3,c2 +Ac4 C Z[i]}
replacing q with q
m2,.,
MAXIMALITY
The possibilities of the ad hoc methods in the proof of Proposition 3.10 are somewhat limited. It is clearly desirable to have algorithms for constructing and identifying maximal orders. In the following we shortly describe how the maximality of a given order can be proved in general. A more detailed version of the algorithm can be found in [I1]. Some of the methods therein are implemented in the Magma software [12]. Suppose we are given a central simple algebra A over F and a finitely generated OF-module A < A. Let k be a multiple of d(A) (c.f. Definition 3.6). The following algorithm describes how the maximality of A can be tested (in polynomial time
1111).
is a maximal Z-order of the division algebra H. Proof: Clearly whole algebra H, the s soite and we have seen that N is a ring, so it is an order of H. Furthermore, if A is any order of H, then so is iA as the element X is in the center of H (cf. Proposition 3.2). Therefore it suffices to show that N is a maximal Z[V'I]order. In what follows, we will call rational numbers in the coset (1/2) + Z half-integers. Assume contrariwise that we could extend the order 'H into a larger order R = 'H[q] by adjoining the quaternion q = a +- a2j, where the coefficients at = + + mt,2 2 + mt,3 3, mnt, c Q for all t, t
Qspango
A[Vs]
Z3[V2.]T
m1,f,
nto, Amntj
As the input the algorithm requires two lists. The first list consists of the prime ideals P of OF which divide k. The ideals P of A which contain PA together constitute the second list. Now Ap is a maximal (OF)p-order if P is not contained in the first list (c.f. Proposition 3.3). We are left with the task of verifying the local maximality at the prime ideals P of the first list. By Propositions 3.8 and 3.9 it then suffices to repeat the algorithm below at each P. STEP 1 Is there exactly one prime ideal X of A in the second list such that PA
