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Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras Thomas Unger and Nadya Markin

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Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras Thomas Unger and Nadya Markin 1 Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras Thomas Unger and Nadya Markin Abstract—In the context of space-time block codes (STBCs), construction will always work with F replaced by K. How- the theory of generalized quaternion and biquaternion algebras ever, when D is noncommutative, one should consider D as a (i.e., tensor products of two quaternion algebras) over arbitrary right K-vector space, i.e., elements of D should be multiplied base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For on the right by scalars from K. It should be taken into account 2 base fields relevant to STBCs, these criteria are exploited, via that in both cases dimF D = (dimK D) , resulting in K- Springer’s theorem, to construct several explicit infinite families matrices of smaller size. of (bi-)quaternion division algebras. These can be used to obtain We are only interested in noncommutative division algebras 2 × 2 and 4 × 4 STBCs via the left regular representation. in this paper. In the STBC literature two large classes of Index Terms—Space-time block codes, division algebras, quad- algebras have been studied: cyclic algebras (e.g. in [2]) and ratic forms. crossed-product algebras (e.g. in [3] and [4]). The latter are characterized by requiring that K is a strictly maximal subfield D F I. INTRODUCTION of which is a Galois extension of . Cyclic algebras are crossed-product algebras with cyclic Galois group Gal(K=F ). VER since Alamouti’s work [1] a decade ago, division We refer to [5] for a historical survey of such algebras. E algebras – both commutative (i.e., fields) and noncom- Determining when precisely these algebras are division mutative – have played an increasingly important role in the algebras can be a nontrivial problem. In this paper we discuss construction of space-time block codes (STBCs) for wireless two classes of algebras, namely (generalized) quaternion and communication. biquaternion algebras. They have characterizations in terms Precisely how this is done is explained very thoroughly in of quadratic forms, a fact that has not fully been exploited [2]. From a mathematical point of view, one wants to construct, yet in the coding theory literature. There are particularly nice for a given integer n, a set S ⊂ Mn(F ) of n×n-matrices over quadratic form-theoretic criteria to determine when they are some field F such that the difference of any two elements of division algebras. S has nonzero determinant. Although many examples of quaternion algebras have ap- The field F is usually an extension of the field Q of rational peared in the literature (first and foremost the Alamouti code numbers, containing an appropriate signal constellation. The of course), they often do so disguised as cyclic algebras (of set S ⊂ Mn(F ) can be obtained as follows: choose some which they are special cases), which hides their quadratic division algebra D of dimension n over F (i.e., D is simul- form-theoretic characterization. taneously an F -vector space and a ring) and consider the left Biquaternion algebras are tensor products of two quaternion regular representation algebras (in the older literature, the term also refers to Hamil- λ : D ! End (D); a 7! λ ; ton’s quaternions considered over the field of complex num- F a bers). They are crossed-product algebras with Galois group the where λa is left multiplication by a, λa(x) = ax for all x 2 Klein 4-group (but the converse inclusion is not true), and are D. Here EndF (D) is the algebra of F -linear transformations thus special cases of the algebras in [4]. Biquaternion algebras on D. After a choice of F -basis for D, EndF (D) can be can also be characterized by a quadratic form. In particular it identified with the algebra Mn(F ). In this way we get an is relatively straightforward to determine when such an algebra embedding λ : D,! Mn(F ). Since all nonzero elements is a division algebra (at least over base fields that are relevant of D are invertible, all nonzero matrices in λ(D) will have for STBCs). It is also easy to produce biquaternion division nonzero determinant. The set S can then be chosen to be (a algebras as tensor products of quaternion division algebras, subset of) λ(D). The difference of any two elements of S will addressing an issue raised in [3, p. 3926]. thus have nonzero determinant. Quaternion division algebras can be used to construct 2 × 2 There are some caveats here: for coding purposes it is often STBCs and biquaternion division algebras to construct 4 × 4 more convenient to consider D as a vector space over some STBCs. subfield K, maximal with respect to inclusion (F ⊂ K ⊂ D). This paper is organized as follows: in Section II we present a When D is commutative (in other words a field), the above brief introduction to quadratic forms. In Sections III and IV we respectively present the theory of quaternion and biquaternion T. Unger and N. Markin are with the School of Mathematical Sci- algebras over arbitrary fields. We also give general quadratic ences, University College Dublin, Belfield, Dublin 4, Ireland (e-mail: [email protected] and [email protected]). form theoretic criteria for such algebras to be division algebras. Version of July 1, 2008 In Section V we obtain matrix representations of quaternion 2 n and biquaternion algebras over arbitrary fields by means of for all v = (v1; : : : ; vn) 2 F and scalars a1; : : : ; an 2 F . It the left regular representation. is convenient to employ the short-hand notation Then we shift our attention to fields that are actually used = ha ; : : : ; a i in space-time block coding: number fields and their transcen- 1 n dental extensions. Such fields can be embedded into bigger to indicate that the matrix of has diagonal entries a1; : : : ; an fields that are complete with respect to a discrete valuation, and zeroes everywhere else. Thus we can write or CDV fields for short. For CDV fields the quadratic form ' ' ha ; : : : ; a i: theoretic criterium for a (bi-)quaternion algebra to be division 1 n can be tested in a finite number of steps by means of Springer’s It is customary to consider quadratic forms up to isometry. As theorem (Section VI). a consequence the entries in ha1; : : : ; ani may be permuted In the final two sections we give examples of infinite fam- and changed by nonzero squares (i.e., any ai may be replaced 2 × ilies of non-isomorphic quaternion and biquaternion algebras by α ai for any α 2 F ). over suitable fields, exploiting Springer’s theorem. Together The dimension of ' is the integer n (i.e., the dimension of with the matrix representations from Section V these can be the underlying vector space). If '0 is another quadratic form 0 used for the explicit construction of 2 × 2 and 4 × 4 STBCs. over F , say of dimension m, ' ' hb1; : : : ; bmi, then the The Appendices contains preliminaries from discrete valu- orthogonal sum of ' and '0, defined by ation theory and number theory. 0 ' ? ' := ha1; : : : ; an; b1; : : : ; bmi n+m II. QUADRATIC FORMS is a quadratic form on F which is uniquely defined up to isometry. For the convenience of the reader we recall some basic facts The quadratic forms ' in this paper will always be from quadratic form theory. We refer to [6] for the details. Let nonsingular in the sense that their determinant det ' := F be a field and write F × := F n f0g. We make the technical a a ··· a 2 F=F 2 is always nonzero. assumption that the characteristic of F is different from 2 (in 1 2 n A quadratic form ' on V is called isotropic over F if other words that 2 is invertible in F ). The fields considered there exists a nonzero vector v 2 V such that '(v) = 0 in this paper are all extensions of the rational numbers and Q and anisotropic over F otherwise. thus will satisfy this condition (but the finite field does not, F2 Note that an anisotropic form over F may become isotropic for example). after scalar extension to a bigger field. Let V be a finite dimensional F -vector space. Remark 2.2: Let ' = ha ; : : : ; a i. If there are Definition 2.1: A quadratic form on V (over F ) is a map 1 n b ; b ; : : : ; b 2 F × such that ' ' h1; −1; b ; b ; : : : ; b i, then ' : V ! F satisfying the following two properties: 3 4 n 3 4 n ' is clearly isotropic. The converse is also true. (i) '(αv) = α2'(v) for all v 2 V and all α 2 F ; (ii) the map b' : V × V ! F defined by III. QUATERNION ALGEBRAS × b'(v; w) := '(v + w) − '(v) − '(w) Definition 3.1: Let a; b 2 F .A (generalized) quaternion algebra over F is a 4-dimensional F -vector space with F - for all v; w 2 V is bilinear (and automatically symmet- basis B = f1; i; j; kg, whose ring structure is determined by ric). 2 2 n i = a; j = b and ij = −ji = k: (2) Concretely, if we identify V with F , where n = dimF V , then ' is just a homogeneous polynomial of degree two: for We denote this algebra by (a; b)F . Its elements are called 1 ≤ i; j ≤ n there exist scalars aij 2 F such that quaternions. 2 n It follows from the defining relations (2) that k = −ab and X '(v) = '(v1; : : : ; vn) = aijvivj; (1) that i, j and k anti-commute.
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