Division Algebras and Wireless Communication B. A. Sethuraman

he aim of this note is to bring to the at- algebras that are finite-dimensional as such vector tention of a wide mathematical audience spaces. Commutative fields are trivial examples the recent application of alge- of these division algebras, but they are by no bras to wireless communication. The means the only ones: for instance, class-field application occurs in the context of theory tells us that over any algebraic Tcommunication involving multiple transmit and field K, there is a rich supply of noncommutative receive antennas, a context known in engineer- division algebras whose center is K and are finite- ing as MIMO—short for multiple input, multiple dimensional over K. output. While the use of multiple receive anten- Interest in MIMO communication began with the nas goes back to the time of Marconi, the basic papers [21, 10, 24, 11], in which it was established theoretical framework for communication using that MIMO wireless transmission could be used multiple transmit antennas was only published both to decrease the probability of error and to about ten years ago. The progress in the field has increase the amount of information that can be been quite rapid, however, and MIMO communica- transmitted. This result caught the attention of tion is widely credited with being one of the key telecommunication operators, particularly since emerging areas in telecommunication. Our focus MIMO communication does not require additional here will be on one aspect of this subject: the resources in the form of either a larger slice of the formatting of transmit information for optimum radio spectrum or increased transmitted power. reliability. The basic setup is as follows: Complex ıφ Recall that a division algebra is an (associative) Re , encoded as the amplitude (R) and phase (φ) algebra with a multiplicative identity in which of a radio wave, are sent from t transmit antennas every nonzero element is invertible. The center of (one number from each antenna), and the encoded a division algebra is the set of elements in the signals are then received by r receive antennas. algebra that commute with every other element in The presence of obstacles in the environment, such the algebra; the center is itself just a commutative as buildings, causes attenuation of the signals; in field, and the division algebra is naturally a vector addition, the signals are reflected several times space over its center. We consider only division and interfere with one another. The combined degradation of the signals is commonly referred to B. A. Sethuraman is professor of at Cali- as fading, and achieving reliable communication fornia State University Northridge. His email address is in the presence of fading has been the most [email protected]. challenging aspect of wireless communication. The author is supported in part by NSF grant DMS- The received and transmitted signals are modeled 0700904. The author wishes to thank P. Vijay Kumar by the relation for innumerable discussions during the preparation of Yr 1 θHr t Xt 1 Wr 1 this article; his counsel was invaluable, his patience mon- × = × × + × where X is a t 1 vector of information signals, umental. The author also wishes to thank Frederique × Oggier for her careful reading of a preliminary version of Y is an r 1 vector of received signals, W is an × this article. r 1 vector of additive noise, H is an r t matrix × ×

1432 Notices of the AMS Volume 57, Number 11 that models the fading, and θ is a in Sk. Often itself is referred to as the space- chosen to multiply the information signals so as time code. ItX is typically assumed that the map to fit the power available for transmission. Under X is “linear in Sk”, that is, it is the restriction k k the most commonly adopted model, the entries to S of a group homomorphism S Mn(C), of the noise vector W and the channel matrix where S is the additive subgroup of C generated→ H are assumed to be Gaussian complex random by S. (The term “space-time” refers to the fact variables that are independent and identically that information (s1, s2, . . . , sk) is packaged in the distributed with zero mean. (A Gaussian complex spatial direction by sending it out through several random variable is one of the form w x ıy, physically separated transmit antennas and in the where x and y are real Gaussian random= variables+ time direction by sending it out in n consecutive that are independent and have the same mean and transmissions.) variance. The modulus of such a random variable, Under the information-theoretic framework de- and in particular the magnitude of each fading veloped by Shannon in 1948 ([18]) and adopted coefficient hij , is then Rayleigh distributed. This ever since within the telecommunication commu- model is hence also known as the Rayleigh fading nity, the amount of information conveyed by a k channel model.) It is the presence of fading in the message in this setting is equal to log2(q ) “bits”. channel that distinguishes this model from more Since this amount of information is conveyed classical channels, where the primary source of in n transmissions over the MIMO channel, the disturbance is the additive Gaussian noise W . rate of information transmission is then given by A common engineering model is to assume k n log2(q) bits per channel use. When q and n are that the channel characteristics (i.e., the fading fixed a priori, the quantity k serves as a measure coefficients hij ) stay constant in some fixed but of information rate. small time interval and that these characteristics Reliability of communication is commonly mea- are known to the receiver but not the transmit- sured by the probability Pe of incorrectly decoding ter. (This is known as coherent transmission.) If the transmitted message at the receiver. The pair- each antenna can transmit n times during such an wise error probability Pe(i, j) (for i ≠ j) is the interval, then the transmissionprocess is compart- probability that message i is transmitted and mes- mentalized into blocks of length n: each antenna sage j is decoded. Performance analysis of MIMO transmits n times, and each receiver waits to re- communication systems typically focuses on the ceive all n transmissions before processing them. pairwise error probability, as it is easier to es- A common simplifying assumption is to take timate and also because the error probability Pe r t n, and the equation above is accordingly can be upper and lower bounded in terms of the = = modified to read pairwise error probability. It was shown in [21, 11] that for a fixed SNR (1) Yn n θHn nXn n Wn n. × = × × + × (i.e., power) ρ, in order to keep the pairwise error Thus the ith column of Y , θX, and W represent probability low, the space-time code must meet (respectively) the received vectors, the transmitted the two criteria below, of which the firstX is primary: information, and the additive noise from the ith (1) Rank Criterion: For s : (s1, s2, . . . , sk) and transmission. A measure of the power available = s′ : (s1′ , s2′ , . . . , sk′ ) with s ≠ s′, the differ- during a single transmission from all n antennas, ence= matrix i.e., a single use of the telecommunication chan- X(s) X(s′) nel, is the signal-to-noise ratio (SNR) ρ. Recall that − must have full rank n, i.e., it must be the Frobenius norm X F of X (xi,j ) equals || || = 2 invertible. i,j xi,j . Since the power required to send a | | (2) Coding Gain Criterion: For s and s′ as complex number varies as the square of its mod- above, s ≠ s′, the modulus of the determi- ulus, the normalization constant θ must satisfy 2 2 nant of difference θ X F nρ. || || ≤ det(θX(s) θX(s′)) A subset S of the nonzero complex numbers | − | known as the signal set is selected as the alphabet must be as large as possible. (a common situation is that S is a finite subset of Clearly, the second criterion comes into play only size q of the nonzero Gaussian Z[ı] 0 ), when the first criterion has been met but then −{ } and a k-tuple (s1, s2, . . . , sk), si S, constitutes the subsumes it. Each criterion impacts a different message that the transmitter wishes∈ to convey to communication parameter, and the two are hence the receiver. Thus there are qk messages in all, and stated independently. Note that one cannot arbi- it is assumed that each message is equally likely trarily scale the matrices X to increase the coding to be transmitted. A space-time code is then a one- gain because the assumption of fixed ρ, along with k k 2 2 to-one map X : S Mn(C); we write for X(S ). the relation θ X nρ, would simply cause a → X || ||F ≤ The transmitted matrix θXn n in Equation (1) is corresponding decrease in θ. Note too that one × thus drawn from the set θ as (s1, s2, . . . , sk) vary cannot increase the quantity k (a proxy for the rate X

December 2010 Notices of the AMS 1433 n of information) arbitrarily, as this would create k 2 for n > 4. It follows that these general- a larger set of matrices θX all circumscribed to izations≤ of the Alamouti code transmit too few lie within a sphere of radius √nρ, which would information symbols for more than two transmit

then cause the determinant of their differences antennas. (A similar analysis of the matrices Ai us- to get smaller, thereby going against the second ing representations of Clifford algebras was made criterion. by Tirkkonen and Hottinen in [20].) In 2001 Sundar Rajan, a professor of com- Satisfying the Rank Criterion munication engineering at the Indian Institute of The earliest space-time code, for two antennas, Science, introduced the problem of designing ma- was given by an engineer, Alamouti ([1]): given an trices X(s1, . . . , sk) satisfying the rank criterion 2 arbitrary signal set S, he chose X : S M2(C) to → to this author. Given his algebraic background, be this author could recognize easily that matrices s1 s2 arising from embeddings of fields and division (2) X(s1, s2) − = s2 s1 algebras can be utilized to solve this problem. Let f : D Mn(C) be an embedding, i.e., an (injective) (where si stands for complex conjugation). Itis easy → to see that the rank criterion is immediately met. ring homomorphism of a division algebra D into C Writing s1 u1 ıu2, s2 u3 ıu4, eachsuch matrix the n n matrices over . Then for X1 f (d1) = + = + 4 × = and X2 f (d2) (X1 ≠ X2), X1 X2 must neces- can be expressed in the form X(s1, s2) i 1 ui Ai . = = = − The 2 2 complex matrices A are such that for sarily be invertible. This is because d1 d2, being j − any complex× 2 2 channel matrix H, the collection a nonzero element of the division algebra D, is × of 2 2 matrices HAi is pairwise orthogonal automatically invertible, and since f is a homo- × { } 8 when regarded as vectors in R by writing out morphism, the same must also be true of X1 X2. − sequentially the real and imaginary parts of each Thus the matrices in f (D) automatically satisfy entry of the HAi . The expansion above makes the rank criterion. Using this observation, Sun- { } it possible to do a least squares estimation of the dar Rajan, his Ph.D. student Shashidhar, and this uj from the received matrix Y , also considered as author ([19]) proposed several schemes for con- 8 a vector in R as above, by projecting onto the structing space-time codes from various signal respective matrices HAj (we will consider this in sets. For each signal set S and for each n, they more detail later). It is this property that makes constructed suitable division algebras D, suitable the Alamouti code so easy to decode, and, not embeddings f : D Mn(C), and suitable injective → surprisingly, the code has since been adopted maps X : Sk f (D), for suitable k. into the IEEE 802.11n “Wireless LAN” standard. In → For simplicity of construction in the noncom- applications, the s , s are typically drawn from 1 2 mutative case, the authors of [19] used cyclic a subset of Z[ı] {Z[ı]. } division algebras for their codes. A cyclic divi- Alamouti’s code× led to a furious search among engineers and coding theorists for generalizations sion algebra is constructed from two data: a field for higher numbers of antennas. Much of the early extension K/F of degree n that is Galois with cyclic Galois group σ , and a nonzero element work (see[22], for example)focused on combinato- rialmethods. The matrix X inEquation (2) isalmost γ F that satisfies the property that for any ∈ i 1 unitary: it satisfies XX (s s s s )I , where the i 1, . . . , n 1, γ is not a norm from K to F. As † 1 1 2 2 2 = − superscript stands for= transpose+ conjugate, and a K-, the algebra is expressible as † I2 stands for the 2 2 identity matrix. Not surpris- n 1 × − ingly, early workers (see [22], for example) sought D Kui n n matrices X(s1, . . . , sk) whose entries come = i 0 × = from the set sj , sj , ısj , ısj , j 1, . . . , k and satisfy {± ± ± ± = } where u is a symbol. The multiplication in this algebra is given by the relations uk σ (k)u (3) XX† (s1s1 sksk)In. = = + + for all k K, and un γ. The bilinearity of ∈ = This quickly leads to a necessary condition: the multiplication, along with these relations, then existence of 2k 1 complex n n matrices Ai allows us to determine the product of any two − × satisfying Ai†Ai In, Ai† Ai , and Ai Aj Aj Ai elements of D. One can prove that this construction for 1 i < j =2k 1.= These − are, of course,= − the ≤ ≤ − indeedyields a divisionalgebra withcenter F. (Such Hurwitz-Radon-Eckmann matrices, and classical a division algebra is said to be of index n.) results of Hurwitz-Radon-Eckmann (see [6], for There is a well-known embedding of such a D instance) severely limits the values of k for which n 1 into Mn(K) that sends k0 k1u kn 1u − to such matrices can exist. If n 2a(2b 1), then + + − the Hurwitz-Radon-Eckmann result= says+ that the maximum possible value of k equals (a 1). Thus 1This is a sufficient condition to obtain a cyclic division 3n + k n if and only if n 2, k for n > 2, and algebra. = = ≤ 4

1434 Notices of the AMS Volume 57, Number 11 The net result, as can easily be seen, is that the 2 n 1 k0 γσ (kn 1) γσ (kn 2) . . . γσ − (k1) determinant of the difference of any two such − 2 − n 1 k1 σ (k0) γσ (kn 1) . . . γσ − (k2)  2 − n 1  matrices will live in Z[ı] and therefore will have k2 σ (k1) σ (k0) . . . γσ − (k3)  2 n 1  modulus bounded below by 1. Moreover, this will  k3 σ (k2) σ (k1) . . . γσ − (k4)  (4)    . . . . .  be true no matter how large a subset of Z[ı] is used  . . . . .   . . . . .    as the signal set. They called this last property  k σ (k ) σ 2(k ) . . . γσ n 1(k )   n 2 n 3 n 4 − n 1   − − 2 − n 1 −  the “nonvanishing determinant property”, and  kn 1 σ (kn 2) σ (kn 3) . . . σ − (k0)   − − −  they called the specific code they proposed the By taking F to be various subfields of C containing Golden Code. It was so named for the Golden Q(S) (the field generated by the elements of S Ratio that appears naturally: it is derived from over Q) in this formulation, and for each such F the division algebra (Q(ı, √5)/Q(ı), σ , ı). Here, σ taking various K and γ, a wide variety of space- is the automorphism of K Q(ı, √5) that sends time codes can be constructed for a wide range of √5 to √5 and acts as the= identity on Q(ı).A signal sets. For further simplicity of construction, − 1 √5 Z[ı]-basis for is given by 1 and φ + . Write particularly in the selection of the element γ above, K 2 O1 √5 = the authors of [19] chose all their base fields F to ψ for σ (φ) − . For asignal set S Z[ı] Q(ı) = 2 ⊂ ⊂ contain transcendental elements; in most cases, (the most common kind of signal set), this code 4 their cyclic extensions K/F were of the form sends S to Mn(C) via the matrix K0(x)/F0(x), where K0/F0 is a cyclic extension of 1 s0,1α s0,2αφ ı(s1,1θ s1,2θψ) number fields, and x is a transcendental. In these (5) + + . √5 s1,1α s1,2αφ s0,1θ s0,2θψ cases, the authors’ construction yielded codes + + n2 C 2 1 X : S Mn( ), i.e., with k n . Here, the scale factor, α 1 ı(1 φ), Alamouti’s→ original code= above arises as a √5 = + − and θ σ (α) 1 ı(1 ψ) are used to shape special case of this formulation: the matrices = = + − of Equation (2) are just the matrices of Equa- the code (more on this later). Comparing with tion (4) above specialized to the cyclic algebra the matrix (4) above and ignoring the scale fac- (C/R, σ , 1), where σ stands for complex con- tor, we see that k0 s0,1α s0,2αφ and k1 − = + = jugation. This is nothing other than Hamilton’s s1,1α s1,2αφ. Note that this code encodes four + : the four-dimensional R algebra R information symbols in each matrix. (A variant Rı Rj Rk subject to the relations ı2 j2 1,⊕ of this code, also based on the division algebra ıj ⊕ jı⊕ k. (The signal set in Alamouti’s= = con- − (Q(ı, √5)/Q(ı), σ , ı), also incorporating the shap- struction= − = is contained in K instead of F, unless of ing criterion described later, is currently part of course S is real.) the IEEE 802.16e “WiMAX” standard. The Alamouti code based on the quaternions is also part of this Satisfying the Coding Gain Criterion standard.) The coding community immediately recognized With the introduction of cyclic division algebras the potential of cyclic division algebras as a funda- as a fundamental construction paradigm and with mental tool for constructing space-time codes and the use of codes constructed with entries from K for suitable extensions of Q(ı), the subject of began to work with the coding paradigm intro- O duced in [19]. However, there was still a drawback. space-time coding took off. It is harmless and very Although the specific codes of [19] certainly sat- often actually useful to assume that the signal set isfied the rank criterion, their performance was S is infinite: typically, S is assumed to be one of not satisfactory. The reason for this became clear: the standard lattices Z, Z[ı] or the Eisenstein lat- the specific division algebras of [19] were pro- tice Z[ω], where ω stands for the primitive third 1 √ 3 − + − posed only for mathematical simplicity—merely root of unity 2 . (Under these assumptions as easy examples of the larger paradigm of di- the code forms an additive group, so one only vision algebras—and were not optimized for the needs to consider the rank of X(s1, . . . , sk) and the coding gain performance criterion above. The use modulus of the determinant det θX(s1, . . . , sk) in | | of transcendental numbers in the codes in [19] the rank criterion and the coding gain criterion.) caused the determinants of the difference matri- Coding theorists immediately looked for specific ces to come arbitrarily close to zero and limited constructions of division algebras of the form their performance. (K/F, σ , γ) for the cases where F Q, F Q(ı), = = This situation was quickly remedied in [2] by a and F Q(√ 3), corresponding to signal sets = − very clever technique. To provide a lower bound on equaling one of the three lattices above. While the moduli of the determinants of the difference such constructions have been known in principle of code matrices, the authors Belfiore, Rekaya, to mathematicians working with division alge- and Viterbo first constructed division algebras bras, the coding theorists absorbed the necessary from cyclic extensions K/Q(ı) and γ Z[ı], but number-theoretic background in very short order ∈ then restricted the various ki in the matrix (4) and explicitly constructed division algebras over above to entries in K , the ring of integers of K. such fields for all indices n ([15] and [7]). (The O

December 2010 Notices of the AMS 1435 hard task here is to select γ F so that it has Themathematicsneededfortheworkonperfect ∈ O the property that γi is not a norm from K to codes is quite interesting. Analyzing the condition

F for i 1, . . . , n 1.) In all such cases, an F - that M be unitary, we find that it is sufficient to = − O basis βj of K is chosen, and each ki is written make the following matrix unitary: n O 2 as j 1 si,j βj for si,j in the signal set. Thus n β β = 1 n elements from the signal set are coded in each ma- σ (β ) σ (β )  1 n  trix, and by construction, the determinant of each U( β , . . . , β ) . 1 n . { } =  .  matrix is nonzero and lies in one of the discrete    σ n 1(β ) σ n 1(β )  lattices above. The modulus of the determinant  − 1 − n    will therefore be bounded below by the length of Here, it is not necessary that the βj be an F basis O the shortest vector in the lattice so the code will of K ; it is sufficient that they be an F linearly O O have the nonvanishing determinant property. independent subset of K . (So, for example, in the Golden Code (5) above,Oα is chosen so that with Other Performance Measures β1 α and β2 αφ, the matrix = = In parallel, as the subject became better under- α αφ stood, several additional performance criteria θ θψ started to be imposed on codes. In a funda- 1 mental paper ([25]), Zheng and Tse provided a is unitary after being multiplied by the scale precise quantification of the trade-off (known as √5 factor.) So the question is: how to find F sub- the diversity-multiplexing gain or “DMG” trade- O off) between information rate and reliability. They modules of K that satisfy this unitary condition? For n 2b,O it is easy to see that for the field defined numerical measures for each of the ben- = efits and showed that the pair of benefits lie K Q(ζ) and F Q(ı), where ζ is a primitive b =2 = in a region of the first quadrant whose upper 2 + th root of unity, the various powers of ζ are Z boundary is a piecewise linear concave up curve. [ı]-linearly independent and satisfy the unitary condition above. For odd n, Elia and coworkers use In the paper [7], Vijay Kumar and his students a construction due to B. Erez [9] that was needed showed that all codes constructed from cyclic di- in a different context: Erez was showing that for vision algebras with the additional nonvanishing certain cyclic extensions K/Q with Galois group determinant property will automatically perform G, the square root of the inverse different is a free at the upper boundary of this region and will Z[G] module that has an orthogonal basis with hence be “DMG optimal”. This of course further respect to the usual trace form on K that sends cemented the use of cyclic division algebras for x, y to T r Q(xy). code construction. K/ The most recent performance criteria for space- Another set of criteria was proposed by Oggier time codes, and in some sense the most mathe- and coworkers in the paper [17]. One first rewrites matically exciting, have come from Lahtonen and the matrix (4) as a single n2 1 vector. When n × coworkers ([13]). For the usual cases in which S ki j 1 si,j βj for si,j in the signal set and βj is one of Z, Z[ı], Z[ω], it is easy to see from the = = 2 an Fbasis for K , this n 1 vector can be ex- linearity of the code matrices X that on writing O O × 2 2 pressed as M.v, where M is an n n matrix and v each X as an n2 1 vector as above and separating × T × is the column vector (s0,1, s0,2, . . . , si,j , . . . , sn 1,n) . the real and imaginary parts, one gets a full lattice − One now requires that the matrix M be unitary in R2n2 , i.e., the additive group generated by 2n2 and that γ 1. The first condition is called good 2n2 | | = linearly independent vectors in R . We refer to shaping, and the idea behind it is that this forces this lattice as the code lattice. After normalizing the average energy needed to send the vector v all code matrices so that infX det(X) 1, they without coding to be the same as that needed to postulate that codes whose lattice∈X | points| = are the send it in the coded matrix form (4). The condition most dense in R2n2 will have the best performance, γ 1 causes the average energy transmitted per | | = and, indeed, they find this is borne out in several antenna to be equal for all transmissions. Oggier circumstances by simulations. To obtain a suitable and coworkers called such codes “perfect” and numerical measure for the relative density, they constructed perfect codes for n 2, 3, 4, and 6. invert the situation: they normalize the code lat- = This was followed by work of Elia and cowork- tice to have fundamental volume 1 instead. Thus ers ([8]), who constructed perfect codes for all they define the normalized minimum determinant values of n and additionally showed that perfect of a code lattice of rank 2n2 in a Q(ı) divi- codes satisfy other information-theoretic proper- sion algebra of index n (embedded in Mn(C)) as ties such as information-losslessness (a concept the minimum of theΛ moduli of the determinants introduced by Damen and coworkers in [4]) and det(X(s1, . . . , sn2 )) as X(s1, . . . , sn2 ) runs through approximate universality (a concept introduced by |the lattice, divided| by the fundamental volume of Tavildar and Viswanath in [23]). . Since a smaller fundamental volume represents

Λ 1436 Notices of the AMS Volume 57, Number 11 a higher density, the goal is to construct codes applies to codes from lattices from maximal or- whose code lattice would maximize this ratio ders and provides a further performance boost in among all full lattices in the division algebra. such cases. Recall that if D isΛ a division algebra with center F and if R is a subring of F whose quotient field Key Challenge: Decoding is F, then an R-order in D is a subring T of D What are some of the key problems that need containing R that is finitely generated as an R- to be solved in space-time codes? Perhaps the module and satisfies TF D. A maximal R-order biggest engineering challenge in the subject is the = is one that is maximal with respect to inclusion. issue of decoding. The problem quite simply is In the typical situation where S is one of Z, Z[ı], the following: given the received vectors in Y (see or Z[ω], so F is one of Q, Q(ı), or Q(√ 3), and Equation (1)), determine the entries of the matrix − where the ki of the matrices in (4) are constrained X that represent the original information. Assume to lie in K and γ F , the code matrices of (4) O ∈ O that k symbols are coded in the matrix X and that naturally form an S-order. Thus the code matrices the entries of X are linear in the signal entries have a dual structure of an S-order and a full lattice s1, ... , sk (typically arising from Z, Z[ı], or Z[ω]). 2n2 in R . Lahtonen and coworkers investigate the By writing out sequentially the real and imaginary interplay between these two structures. They ask: parts of each entry of Y , W , and s1, ... , sk, we How will the code’s performance as measured may rewrite Equation (1) as Y Zv W . Here Z by its normalized minimum determinant vary if, is a 2n2 2k real matrix that= depends+ on H, θ, in addition to carrying its natural structure of a × 2 and the parameters of the code matrix X, v is the full Z-lattice in R2n , we choose our code to form T signal vector (x1, y1, . . . , xi , yi , . . . , xk, yk) with xi an arbitrary S-order inside an F-division algebra? and yi being the real and imaginary parts of si, and In these cases, the minimum modulus of the similarly for Y and W . If the columns of Z were determinantsof the codematricesis1, soitfollows orthonormal, decoding would be quite simple: we from the definition of the normalized minimum would have ZT Y v ZT W with ZT W also hav- determinant that the smaller the fundamental ing independent,= identically+ distributed Gaussian volume of the lattice the better the code. If T1 and entries. Hence, under maximum likelihood esti- T are S-orders and and the corresponding 2 T1 T2 mation, v can be taken to be the closest vector lattices with fundamental volumes V and V , T1 T2 in Sk (viewed inside the Euclidean space R2k) to then T1 T2 impliesΛ T Λ T , which in turn ⊆ 1 ⊆ 2 ZT Y . This is a very simple and computationally means that VT VT . It follows therefore that 2 ≤ 1 fast scheme: we march through ZT Y component the best normalized minimumΛ Λ determinant will pair by component pair, and we find the element arise when a maximal order is used for the code. of the signal lattice S closest to that component The authors then relate the fundamental volume pair. of the code lattice to the Z-discriminant of the The process above is called single symbol de- maximal order and then invoke known formulas coding. (For k < n2 this is the same as orthogonal for discriminants of maximal orders to compute 2n2 the best normalized minimum determinant of projection onto the subspace of R determined by the columns of Z.) There are some nice sit- codesarisingfrom F ordersinsideagivendivision algebra. In particular,O they show (for the fields Q(ı), uations in which the matrix Z is (essentially) Q(√ 3) and Q) that the best division algebras to orthogonal; this happens in the case of the use will− be ones that are ramified at precisely two Alamouti code, and more generally, in the codes of the “smallest” primes of the field (where the size satisfying Equation (3). The matrix Z for such codes T 2 of a prime P < π > is defined to be the modulus satisfies ZZ θ T r(HH†)I2k. We may divide the = = π ). Thus, for Q(ı) for example, one needs to relation Y Zv W by θ T r(HH†). The entries of | | = + transmit on a code arising from a maximal order the new noise vector 1/(θ T r(HH†))W are still in- inside a division algebra ramified only at (1 ı) dependent identically distributed Gaussian, while + and (2 ı) (or (2 ı)). (Much of this was part of the columns of the matrix 1/(θ T r(HH†))Z are + − Vehkalahti’s Ph.D. thesis.) now orthonormal. Thus single symbol decoding One of the drawbacks of using maximal orders can be employed in all these cases. is that the corresponding code lattice may not But for other codes Z is rarely orthogonal! have good shape. Thus optimizing a code for min- In general, given that the entries of W are in- imum normalized determinant may destroy any dependent identically distributed Gaussian, for optimization for shape. The recent work of Raj maximum likelihood estimation one needs to 2 Kumar and Caire ([3]) proposes a very clever tech- search in Sn (viewed inside the Euclidean space 2n2 T nique of mapping lattice points to certain cosets R ) for that vector v (x1, y1, . . . , xn2 , yn2 ) such = of a suitably chosen sublattice of a standard cubic that Zv is closest to Y . (Here we will assume lattice; this smooths out an irregular lattice and that k n2, as is usually the case for codes from gives it better shape. In particular, their technique cyclic= division algebras.) This can no longer be

December 2010 Notices of the AMS 1437 accomplished symbol by symbol, and one needs above (modulo the subgroup 1 ) turns out to be 2 {± } to search in the full space Sn instead of just in S. a Kleinian group, i.e., a discrete subgroup of the There is an algorithm called the sphere decoding projective special linear group PSL2(C). As a sub- algorithm (see [5], for instance) that accomplishes group of PSL2(C), U1(R) (modulo 1 ) acts on {± } this search in an intelligent manner, but as is to be the upper half-space model of hyperbolic 3-space expected of any search in Sn2 , even this algorithm H3 as a group of orientation-preserving isometries, gets very cumbersomeonce n exceeds2. (One must and Poincaré’s fundamental polyhedron theorem keep in mind that the search for v such that Zv gives a set of generators and relations for such is closest to Y is essentially a closest lattice point a group in terms of certain automorphisms of a search, and this is known to be NP-hard. What fundamental for the group. Given a point 3 saves the day is that the received vectors Y are not P in H , the Dirichlet polyhedron centered on P random but have a Gaussian distribution about is the closure of the set of points x such that dH (x, P) < dH (g(x), P) for all g U1(R) (modulo the lattice vectors Zv. In [12], Hassibi and Vikalo ∈ show that under certain technical assumptions, 1 ), g ≠ 1, where dH is the hyperbolic metric on {±3 } the expected complexity of the sphere decoding H . The authors construct a Dirichlet polyhedron centered on J (0, 0, 1); this is a fundamental algorithm is polynomial, although the worst-case = complexity is exponential.) domain for U1(R). From this polyhedron, using Since Z is rarelyorthogonal,we mayaskwhether Poincaré’s theorem and a computer search, they determine a set of generators of 1(R). They do we can take advantage of the obvious algebraic U structure of the code and simplify the closest this ahead of time and store the results. Next, in vector problem for our particular application. A real time, given a fading matrix H (normalized to very clever set of ideas of Luzzi et al. [16] does have determinant 1), they need to find an element 1 just that and gives an approximate solution to the U of 1(R) such that the Frobenius norm of UH− U 1 decoding problem for the Golden Code (Equation is minimized. They observe that viewing UH− as 3 (5)) by reducing the situation to the action of an element of PSL2(C) acting on H , the Frobe- 3 1 1 SL2(C) on three-dimensional hyperbolic space H . nius norm of UH− is just 2 cosh dH (J, UH− (J)), Their work is a veritable tour de force of the ap- where J and dH are as above. Since U is an isom- plication of abstract mathematics to engineering etry, they must find U 1(R) that minimizes 1 1 ∈ U problems. Their goal is to approximate the channel cosh dH (U − (J), H− (J)). From the definition of matrix H (normalized to have determinant 1) by Dirichlet polyhedra, this means that they need an element U of determinant 1 in the Z[ı]-order to find a Dirichlet polyhedron centered on some 1 1 R ( K /Z[ı], σ , ı). Writing H EU with E simply U − (J) which contains H− (J). They use the ge- = O 1 = 3 being the error HU − , they argue that choosing U ometry of H relative to the action of U1(R) to 1 1 so that the Frobenius norm of E− UH− is min- find such a U: they just need to repeatedly con- imized approximates the original problem= by the sider the various Dirichlet polyhedra centered on 2 following: given a vector Y in Cn and an unknown J and the various gi (J), where the gi run through 2 n the generators of 1(R) that they have computed vector S in Z[ı] determine a “best” estimate of U S if the difference vector W Y S is known ahead of time, along with their inverses. to be approximately independent= identically− dis- tributed Gaussian (in a suitable sense). Given this Role of Mathematicians assumption about the noise vector W , a reasonable What is the role of mathematicians in this field? way to proceed is to assume that W is actually The subject is clearly very mathematical; yet, un- independent identically distributed Gaussian. In like classical coding theory, which now has a this situation, the maximum-likelihood estimate mathematical life of its own and can, for instance, of S is obtained by taking the ith entry of S to be be thought of as a theory of subspaces of vec- the lattice point in Z[ı] closest to the ith entry of tor spaces over finite fields, the center of gravity Y . The authors find that their scheme gives a fast of space-time codes currently lies very solidly in and acceptably accurate decoding. engineering. There is as yet no deep independent What is fascinating is the mathematics behind “mathematics of space-time codes”: the driving their choice of U. First, they need to determine force behind the subject consists of fundamental generators and relations for the group of norm 1 engineering problems that need to be solved be- units 1(R) of R (i.e., the set of multiplicatively fore MIMO wireless communication reaches its full invertibleU elements of R whose determinant as a practical potential, particularly for three or more code matrix is 1). In general, it is very difficult to antennas. This author therefore believes that, as find these for orders in division algebras,but in the things stand now, isolated mathematical investiga- case of certain special algebras over tions of space-time codes that are not grounded in number fields, generators and relations for 1(R) concrete engineering questions would very likely are known. Much of the idea behind thisU goes lead to sterile results. At least for now, mathe- back to Poincaré. The norm 1 units in the order R maticians can best contribute to the subject by

1438 Notices of the AMS Volume 57, Number 11 working in collaboration with engineers who are [12] B. Hassibi and H. Vikalo, On sphere decoding al- motivated by fundamental engineering questions. gorithm. I. Expected complexity, IEEE Trans. Signal 53 This author has found that the leading engineers Process. , no. 8 (2005), 2806–2818. [13] C. Hollanti, J. Lahtonen, K. Ranto and R. in the field already have a practical and intuitive Vehkalahti, On the densest MIMO lattices from understanding of much abstract mathematics but cyclic division algebras, IEEE Trans. Inform. Theory welcome help from trained mathematicians. (This 55, no. 8 (2009), 3751–3780. author has also found that they are a genuine plea- [14] G. Ivanyos and L. Rónyai, On the complexity of finding maximal orders in algebras over Q, sure to collaborate with.) There is clearly a lot of Computational Complexity 3 (1993), 245–261. work for mathematicians to do: particularly in de- [15] T. Kiran and B. Sundar Rajan, STBC-schemes with coding systems with large numbers of receive and nonvanishing determinant for certain number of transmit antennas, but also in other areas of MIMO transmit antennas, IEEE Trans. Inform. Theory 51, no. 8 (2005), 2984–2992. communication that we have not touched upon in [16] Laura Luzzi, Ghaya Rekaya-Ben Othman, and this article, such as cooperative communication in Jean-Claude Belfiore, Algebraic reduction for networks, or noncoherent communication, where space-time codes based on quaternion algebras, the matrix H is not known to either the receiver arXiv:0809.3365v2[cs.IT]. See also Algebraic Reduc- or the transmitter. tion for the Golden Code, Proc. IEEE International Conference on Communications (ICC 2009). [17] F. Oggier, G. Rekaya, J. C. Belfiore, and E. References Viterbo, Perfect space-time block codes, IEEE [1] S. Alamouti, A transmitter diversity scheme for Trans. Inform. Theory 52, no. 9 (2006), 3885–3902. wireless communications, IEEE J. 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December 2010 Notices of the AMS 1439