The nonassociative algebras used to build fast-decodable space-time block codes

Let $K/F$ and $K/L$ be two cyclic Galois field extensions and $D=(K/F,\sigma,c)$ a cyclic algebra. Given an invertible element $d\in D$, we present three families of unital nonassociative algebras over $L\cap F$ defined on the direct sum of $n$ copies of $D$. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors. We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-$m$ for $nm$ transmit and $m$ receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most $\mathcal{O}(M^{15})$.


Introduction
Space-time block codes (STBCs) are used for reliable high rate transmission over wireless digital channels with multiple antennas at both the transmitter and receiver ends. From the mathematical point of view, a space-time block code is a set of complex n × m matrices, the codebook, that satisfies a number of properties which determine how well the code performs.
Recently, several different constructions of nonassociative algebras appeared in the literature on fast decodable STBCs, cf. for instance Markin and Oggier [6], Srinath and Rajan [16], or [11], [12], [19], [14]. There are two different types of algebras involved. The aim of this paper is to present them in a unified manner and investigate their structure, in order to be able to build the associated (fully diverse, fast-decodable) codes more efficiently in the future.
Let K/L be a cyclic Galois field extension with Galois group Gal(K/L) = τ of degree n and K/F a cyclic Galois field extension with Galois group Gal(K/F ) = σ of degree m.
Put F 0 = F ∩ L. Given the direct sum A of n copies of a cyclic algebra D = (K/F, σ, c), c ∈ F 0 , we define three different multiplications on A, which each turn A into a unital nonassociative algebra over F 0 . We canonically extend τ to an L-linear map τ : D → D, choose an element d ∈ D × and define a multiplication on the right D-module for all x, y ∈ D, 0 ≤ i, j < n. We call the resulting algebra It n (D, τ, d), It n M (D, τ, d) or It n R (D, τ, d), respectively. For A = It n (D, τ, d) and A = It n M (D, τ, d), the left multiplication L x with a non-zero element x ∈ A can be represented by an nm × nm matrix with entries in K (considering A as a right K-vector space of dimension mn).
For d ∈ L × , left multiplication L x with a non-zero element x ∈ A = It n R (D, τ, d) is a K-endomorphism as well, and can be represented by an nm × nm matrix with entries in K.
The family of matrices representing left multiplication in any of the three cases can be used to define a STBC C, which is fully diverse if and only if A is division, and fast-decodable for the right choice of D.
The three algebra constructions in this paper generalize the three types of iterated algebras presented in [12] (the n = 2 case). A first question concerning their existence can be found in Section VI. of [6]; the iterated codes treated there arise from the algebra It 2 (D, τ, d).
The algebras It n (D, τ, d) and It n R (D, τ, d) appear when designing fast-decodable asymmetric multiple input double output (MIDO) codes: It n R (D, τ, d) is implicitly used in [16] but not mentioned there, the algebras It n (D, τ, d) are canonical generalizations of the ones behind the iterated codes of [6], and are employed in [11]. Both times they are used to design fast decodable rate-2 MIDO space-time block codes with n antennas transmitting and 2 antennas receiving the data. All codes for n > 2 transmit antennas presented in [16] and all but one [11] have sparse entries and therefore do not have a high data rate.
We include the third family, It n M (D, τ, d), for completeness. After the preliminaries in Section 2, the algebras It n (D, τ, d) and It n M (D, τ, d) are investigated in Section 3. Several necessary and sufficient conditions for It n (D, τ, d) to be a division algebra are given if d ∈ F × . For instance, if n is prime and in case n = 2, 3, additionally F 0 contains a primitive nth root of unity, then It n (D, τ, d) is a division algebra for all d ∈ F \ F 0 with d m ∈ F 0 (Proposition 13). Section 4 deals with the algebras It n R (D, τ, d) which were defined by B. S. Rajan and L. P. Natarajan (and for d ∈ L \ F yield the codes in [16]). They were already defined previously in a little known paper by Petit [10] using twisted polynomial rings. Necessary and sufficient conditions for It n R (D, τ, d) to be a division algebra are given and simplified for special cases. E.g., if D is a quaternion division algebra, It 3 R (D, τ, d) is a division algebra for all d ∈ L \ F with d ∈ N K/L (K × ) (Theorem 17).
Some of these conditions are simplification of the ones contained in an earlier version of this paper, applied in [11] when designing fully diverse codes. In particular for the case n = 3, Proposition 13 makes it easy now to build fully diverse codes of maximal rate using It 3 (D, τ, d) and Theorem 17 using It 3 R (D, τ, d). Previously, there were no criteria known to check such iterated 6 × 3-codes for full inversibility.
How to design fully diverse fast-decodable multiple input multiple output (MIMO) codes for nm transmit and m receive antennas employing certain It n R (D, τ, d) and It n (D, τ, d) is explained in Sections 5 and 6: if the code associated to D is fast-decodable, then so is the one associated to It n R (D, τ, d), respectively, It n (D, τ, d). We are interested in a high data rate and use the mn 2 degrees of freedom of the channel to transmit mn 2 complex symbols.
Our method yields fully diverse codes of rate-m for nm transmit and m receive antennas, which is maximal rate for m receive antennas. We present two examples of a rate-3 code for 6 transmit and 3 receive antennas which are fast-decodable with ML-decoding complexity at most O(M 15 ) (using the M-HEX constellation). One of them is DMT-optimal and has normalized minimum determinant 49( 2 √ 28E ) 18 = 1/7 7 E 9 . We also give an example of a rate-4 code for 8 transmit and 4 receive antennas which is fast-decodable with ML-decoding complexity at most O(M 26 ) (using the M-QAM constellation). The suggested codes have maximal rate in terms of the number of complex symbols per channel use (cspcu).

Preliminaries
2.1. Nonassociative algebras. Let F be a field. By "F -algebra" we mean a finite dimensional nonassociative algebra over F with unit element 1.
A nonassociative algebra A = 0 is called a division algebra if for any a ∈ A, a = 0, the left multiplication with a, L a (x) = ax, and the right multiplication with a, R a (x) = xa, are bijective. A is a division algebra if and only if A has no zero divisors [17, pp. 15, 16].
For an F -algebra A, associativity in A is measured by the associator [x, y, z] = (xy)z − nucleus is an associative subalgebra of A containing F 1 and x(yz) = (xy)z whenever one of the elements x, y, z is in Nuc(A). The commuter of A is defined as Comm(A) = {x ∈ A | xy = yx for all y ∈ A} and the center of A is C(A) = {x ∈ A | x ∈ Nuc(A) and xy = yx for all y ∈ A}.
For coding purposes, often algebras are considered as a vector space over some subfield K, Usually K is maximal with respect to inclusion. For nonassociative algebras, this is for instance possible if K ⊂ Nuc(A).
If then left multiplication L x is a K-linear map for an algebra A over F we can consider the map , after choosing a K-basis for A and expressing the endomorphism L x in matrix form. For an associative algebra, this is the left regular representation of A.
If A is a division algebra, λ is an embedding of vector spaces.
Similarly, given an associative subalgebra D of A such that A is a free right D-module and such that left multiplication L x is a right D-module endomorphism, we can consider the map

2.2.
Associative and nonassociative cyclic algebras. Let K/F be a cyclic Galois extension of degree m, with Galois group Gal(K/F ) = σ . For any c ∈ K\F , the nonassociative cyclic algebra A = (K/F, σ, c) of degree m is given by the m-dimensional K-vector space A = K ⊕ eK ⊕ e 2 K ⊕ · · · ⊕ e m−1 K together with the rules (e i x)(e j y) = for all x, y ∈ K, 0 ≤ i, j, < m, which are extended linearly to all elements of A to define the multiplication of A.
The unital algebra (K/F, σ, c), c ∈ K \ F is not (n + 1)st power associative, but is built similar to the associative cyclic algebra (K/F, σ, c), where c ∈ F × : we again have xe = eσ(x) and e i e j = c for all integers i, j such that i + j = m, so that e m is well-defined and e m = c. (K/F, σ, c) has nucleus K and center F . If c ∈ K \ F is such that 1, c, c 2 , . . . , c m−1 are linearly independent over F , then A is a division algebra. In particular, if m is prime, then A is division for any choice of c ∈ K \ F . Nonassociative cyclic algebras are studied extensively in [18]. [12]. Let K/F be a cyclic Galois extension of degree m with Galois group Gal(K/F ) = σ and τ ∈ Aut(K). Define L = Fix(τ ) and F 0 = L ∩ F . Let D = (K/F, σ, c) be an associative cyclic algebra over F of degree m. For x = x 0 + ex 1 + e 2 x 2 + · · · + e m−1 x m−1 ∈ D, define the L-linear map τ : D → D via

Iterated algebras
If τ m = id then τ m = id.

(i) A is a division algebra if
(ii) Suppose c ∈ Fix(τ ). Then: 2.4. Design criteria for space-time block codes. A space-time block code (STBC) for an n t transmit antenna MIMO system is a set of complex n t × T matrices, called codebook, that satisfies a number of properties which determine how well the code performs. Here, n t is the number of transmitting antennas, T the number of channels used.
Most of the existing codes are built from cyclic division algebras over number fields F , in particular over F = Q(i) or F = Q(ω) with ω = e 2πi/3 a third root of unity, since these fields are used for the transmission of QAM or HEX constellations, respectively.
One goal is to find fully diverse codebooks C, where the difference of any two code words has full rank, i.e. with det(X − X ′ ) = 0 for all matrices X = X ′ , X, X ′ ∈ C.
If the minimum determinant of the code, defined as is bounded below by a constant, even if the codebook C is infinite, the code C has nonvanishing determinant (NVD). Since our codebooks C are based on the matrix representing left multiplication in an algebra, they are linear and thus their minimum determinant is given by If C is fully diverse, δ(C) defines the coding gain δ(C) 1 n t . The larger δ(C) is, the better the error performance of the code is expected to be.
If a STBC has NVD then it will perform well independently of the constellation size we choose. The NVD property guarantees that a full rate linear STBC has optimal diversitymultiplexing gain trade-off (DMT) and also an asymmetric linear STBC with NVD often has DMT (for results on the relation between NVD and DMT-optimality for asymmetric linear STBCs, cf. for instance [15]).
We look at transmission over a MIMO fading channel with n t = nm transmit and n receive antennas, and assume the channel is coherent, that is the receiver has perfect knowledge of the channel. We consider the rate-n case (where mn 2 symbols are sent). The system is modeled as with Y the complex n r × T matrix consisting of the received signals, S the the complex n t × T codeword matrix, H is the the complex n r × n t channel matrix (which we assume to be known) and N the the complex n r × T noise matrix, their entries being identically independently distributed Gaussian random variables with mean zero and variance one. ρ is the average signal to noise ratio.
Since we assume the channel is coherent, ML-decoding can be obtained via sphere decoding. The hope is to find codes which are easy to decode with a sphere decoder, i.e. which are fast-decodable: Let M be the size of a complex constellation of coding symbols and assume the code C encodes s symbols. If the decoding complexity by sphere decoder needs For a matrix B, let B * denote its Hermitian transpose. Consider a code C of rate n. Any X ∈ C ⊂ Mat mn×mn (C) can be written as a linear combination Let S be a real constellation of coding symbols. A STBC with s = nm 2 linear independent real information symbols from S in one code matrix is called l-group decodable, if there is a partition of {1, . . . , s} into l nonempty subsets Γ 1 , . . . , Γ l , so that M g,k = 0, where g, k lie in disjoint subsets Γ i , . . . , Γ j . The code C then has decoding complexity O(|S| L ), where

General iteration processes I and II
We will use the notation defined below throughout the remainder of the paper: Let F and L be fields and let K be a cyclic extension of both F and L such that (3) σ and τ commute: στ = τ σ.
of order n, see the definition of τ in Section 2.3.
for all x, y ∈ D, i, j < n, and call the resulting algebra It n (D, τ, d), or via for all x, y ∈ D, i, j < n, and call the resulting algebra It n M (D, τ, d).
It n (D, τ, d) and It n M (D, τ, d) are both nonassociative algebras over F 0 of dimension nm 2 [F : F 0 ] with unit element 1 ∈ D and contain D as a subalgebra. For both, are the iterated algebras from Section 2.3. The algebras It n (D, τ, d) are canonical generalizations of the ones behind the iterated codes of [6], and employed in [11].
Mat m (K) as a subalgebra and has zero divisors.
Proof. (i) Restricting the multiplication of A to entries in K proves the assertion immediately: By slight abuse of notation, we have It n (K, τ, d) = (K/L, τ, d).
(ii) is trivial as D ⊗ F0 K ∼ = Mat m (K) splits. If d ∈ L × then A has the F 0 -subalgebra (K/L, τ, d), which as an algebra has splitting field K.
(iii) It is straightforward to check that A is isomorphic to D ⊕ f s D, which is a subalgebra of A under the multiplication inherited from A.
(iv) By linearity of multiplication, we only need to show that for all x, y, z ∈ D and all integers 0 ≤ i, j ≤ n − 1. A straightforward calculation shows that these are equal if and only if τ (x) τ (y) = τ (xy) for all x, y ∈ D. This is true if and only if Lemma 3 (iii) can be generalized to the case where n is any composite number if needed.
A is a free right D-module of rank n, with right D-basis {1, f, . . . , f n−1 } and we can embed that we obtain a well-defined additive map If we represent y as a column vector (y 0 , y 1 , . . . , y n−1 ) T , then we can write the product of x and y in A as a matrix multiplication where M (x) is an n × n matrix with entries in D given by  (ii) It remains to show that λ(M (x)) is invertible for every nonzero x ∈ A implies that A is division: for all x = 0, y = 0 we have that xy = λ(M (x))y = 0 implies that y = λ(M (x)) −1 0 = 0, a contradiction.
The following result concerning left zero divisors is proved analogously to [16], Appendix A and requires Lemma 8: 3.1. In this section, A = It n (D, τ, d). We assume d ∈ F × , unless explicitly stated otherwise. (ii) Let F ′ and L ′ be fields and let K ′ be a cyclic extension of both F ′ and L ′ such that Proof. (i) follows from Theorem 9 [10, (2)].
(ii) follows from (i), since every isomorphism preserves the middle nucleus.
for all x, y ∈ D, i, j < n, and call the resulting algebra It n R (D, τ, d). (ii) Let F ′ and L ′ be fields and let K ′ be a cyclic extension of both F ′ and L ′ such that (iii) If d ∈ K × , then the (associative or nonassociative) cyclic algebra (K/L, τ, d) of degree n, viewed as algebra over F 0 , is a subalgebra of A.
(iv) For n > 3, n even, It R (D, τ, d) is isomorphic to a proper subalgebra of It n R (D, τ, d).
as subalgebra and has zero divisors.
The proofs of (i) and (ii) are analogous to the one of Lemma 8, the ones of (iii), (iv), (v) to the ones in Lemma 3. for all z ∈ D.

(iii) (cf. [2]) It 4 R (D, τ, d) is a division algebra if and only if
and (iv) Suppose that n is prime and in case n = 2, 3, additionally that F 0 contains a primitive [13], the assertion now follows as in the proof of Theorem 9.
Proposition 16. Suppose that n is prime and in case n = 2, 3, additionally that F 0 contains a primitive nth root of unity.
(ii) The proof is a straightforward calculation analogous to (i) or the proof of [16, Proposition 5].

4.2.
A = It n R (D, τ, d) is a right K-vector space of dimension mn. By choosing d ∈ L × from now on, we achieve that left multiplication L x is a K-endomorphism and can be represented by a matrix with entries in K.
For d ∈ L × , the algebras A = It n R (D, τ, d) are behind the codes defined by Srinath and Rajan [16], even though they are not explicitly defined there as such. In the setup of [16], it is assumed that d ∈ L \ F and that L = F . We do not assume that L = F for now.
Example 18. Let F 0 have characteristic not 2 and D = (K/F, σ, c) = K ⊕ eK be a quaternion division algebra over F with multiplication Let K/L be a quadratic field extension with non-trivial automorphism τ , Here, xu is given in equation (6), Thus we can write the multiplication in terms of the K-basis {1, e, f, f e} as Since d ∈ K, it commutes with the elements y i and v i in the above expression.
Write Φ(x + f y) for the column vector with respect to the K-basis, i.e., then we can write the product as The matrix on the left side is equal to λ(x) dλ( τ (y)) λ(y) λ( τ (x)) .
Thus for d ∈ L × , left multiplication L x is a K-endomorphism and can be represented by the above matrix with entries in K.
In the following, A = It n R (D, τ, d) and we assume d ∈ L × . Any element x ∈ A can be identified with a unique column vector Φ(x) ∈ K mn using the standard K-basis  (iii) For every x ∈ A, det(λ(M (x))) ∈ L.
Proof. (i) For any r ∈ D with r = r 0 + er 1 + · · · + er m−1 , where r 0 , . . . , r m−1 ∈ K, define For any x i and y i , the multiplication in D is given by φ(x i y i ) = λ(x i )φ(y i ).
Moreover, since d ∈ L we see that φ(d) = [d, 0, . . . , 0], and therefore Now it is straightforward to see that the matrix multiplication Λ(x)Φ(y) does indeed represent the multiplication in A. (iii) It enough to show that det(Λ(x)) = τ (det(Λ(x))) = det(τ (Λ(x))), where It follows that Λ(x) = P τ (Λ(x))P −1 , with where I m is the m × m identity matrix and 0 is the m × m zero matrix which proves the assertion.

5.
How to design fast-decodable Space-Time Block Codes using It n R (D, τ, d)

To construct fully diverse space-time block codes for mn transmit antennas using
It n R (D, τ, d) (or It n (D, τ, d) in the next Section), let L be either Q(i) or Q(ω), ω = e 2πi/3 , and D = (K/F, σ, c) a cyclic division algebra of degree m over a number field F = L, c ∈ F ∩ L, and where K is a cyclic extension of L of degree n with Galois group generated by τ . We assume that σ and τ commute. For x ∈ D, let λ(x) be the m × m matrix with entries in K given by the left regular representation in D.
Each entry of λ(x) can be viewed as a linear combination of n independent elements of L. As such we express each entry of these as a linear combination of some chosen L-basis Thus an entry λ(x) has the form The elements s i , 1 ≤ i ≤ mn, are the complex information symbols with values from QAM (Z(i)) or HEX (Z(ω)) constellations.

5.2.
We assume that f (t) = t n −d ∈ D[t; τ −1 ], d ∈ L × , is irreducible. Then A = It n R (D, τ, d) is division and each codeword in C is a matrix of the form given in (7) and these are invertible mn × mn matrices with entries in K.
Contrary to [16], we are interested in high data rate, i.e. we use the mn 2 degrees of freedom of the channel to transmit mn 2 complex information symbols per codeword. If mn channels are used the space-time block code C consisting of matrices S of the form (7) with entries as in (8)  is l-group decodable, then C has ML-decoding complexity O(M mn 2 −mn(l−1)/l) ) and is fastdecodable.
We use M -HEX complex constellations and the notation from 5.1 (i.e., s j ∈ Z[ω]): choose {θ 1 , θ 2 , θ 3 } to be a basis of the principal ideal in O K generated by θ 1 with θ 1 = 1 + ω + θ, Each codeword S(λ(x 0 )) = diag[λ(x 0 ), τ (λ(x 0 )), τ 2 (λ(x 0 ))] is 2-group decodable [16,Proposition 7]. S(λ(x 0 )), S(λ(x 1 )) and S(λ(x 2 )) contain each 6 complex information symbols. By Proposition 20, the ML-decoding complexity of the code is at most O(M 15 ) and the code is fast-decodable. We are no experts in coding theory but assume that hard-limiting the code as done in [16]  The associated code is With the encoding from 5.1, we encode 32 complex information symbols with each codeword S. The code has rate 4 for 8 transmit and 4 receive antennas which is maximal. Assuming We have i = z τ (z) τ 2 (z) τ 3 (z) for any z ∈ D [16]. We are not able to check whether the code is fully diverse, since we cannot exclude the possibility that F (t) = t 4 − i decomposes into two irreducible polynomials in D[t; τ −1 ], we are only able to exclude some obvious cases.
6. How to design fast-decodable fully diverse MIMO systems using It n (D, τ, d) with d ∈ F \ F 0 and n prime We assume the set-up from Section 5.1 with the additional condition that n is prime and in case n = 2, 3, additionally that F 0 contains a primitive nth root of unity. In order to construct fully diverse codes, we do not need to restrict our considerations to sparse codes as done in [11]: 6.1. We assume that d ∈ F \ F 0 , such that d m ∈ F 0 . Then A = It n (D, τ, d) is division and each codeword in C is a matrix of the form given in (1), which becomes (7), as d ∈ F , hence These are invertible mn × mn matrices with entries in K. C is fully diverse by Proposition 13.
Remark 22. Suppose that n is an odd prime. If n = 3, additionally assume that F 0 contains a primitive nth root of unity. Then for d ∈ F = F 0 (α), d = d 0 + d 1 α · · · + d n−1 α n−1 (d i ∈ F 0 ), it is easy to calculate examples with d m ∈ F 0 , e.g. if n > 2 is a prime and m = 2, any d = d 0 + d 1 α, d 1 = 0 works.
Contrary to [11], we now use the mn 2 degrees of freedom of the channel to transmit mn 2 complex information symbols per codeword. If mn channels are used, the space-time block code C consisting of matrices S of the form (7) with entries as in (8) has a rate of n complex symbols per channel use, which is maximal for n receive antennas. By Proposition 20, which holds analogously, if the subset of codewords in C made up of the diagonal block matrix S(λ(x 0 )) = diag[λ(x 0 ), τ (λ(x 0 )) . . . , τ n−1 (λ(x 0 ))] is l-group decodable, then C has ML-decoding complexity O(M mn 2 −mn(l−1)/l) ) and is fastdecodable.
However, NVD seems not always necessary for DMT-optimality to hold.

Conclusion
One current goal in space-time block coding is to construct space-time block codes which are fast-decodable in the sense of [4], [7], [8] also when there are less receive than transmit antennas, support high data rates and have the potential to be systematically built for given numbers of transmit and receive antennas.
After obtaining conditions for the codes associated to the algebras It n (D, τ, d), d ∈ F × , and It n R (D, τ, d), d ∈ L \ F , to be fully diverse, we construct fast decodable fully diverse codes for mn transmit and n receive antennas with maximum rate n out of fast decodable codes associated with central simple division algebras of degree m, for any choice of m and n. We thus answer the question for conditions to construct higher rare codes [16,VII.].
The conditions were simplified in the special case of a quaternion algebra D and an extension K/L with [K : L] = 3 in Theorem 17, yielding an easy way to construct fully diverse rate-3 codes for 6 transmit and 3 receive antennas using It 3 R (D, τ, d), d ∈ L \ F . They were further simplified for prime n if n = 3 or if F 0 contains a primitive nth root of unity (Proposition 13), using It n (D, τ, d), d ∈ F \ F 0 for the code construction.
Since we are dealing with nonassociative algebras and skew polynomial rings, there is no well developed theory of valuations or similar yet which one could use to study the algebras over number fields. This would go beyond the scope of this paper and will be addressed in [2].