Integral Quaternions Research Articles

Encoder Lipschitz integers: the Lipschitz integers that have the “division with small remainder” property

Encoder Lipschitz integers: the Lipschitz integers that have the “division with small remainder” property

Kodlayıcı Hurwitz Tamsayıları: Küçük Kalanlı Bölme Özelliğine Sahip Hurwitz Tamsayıları

Considering error-correcting codes over Hurwitz integers, prime Hurwitz integers are considered. On the other hand, considering transmission over Gaussian channel, Hurwitz integers, whose the norm is either a prime integer or not a prime integer, are considered. In this study, we consider Hurwitz integers, the greatest common divisor of components of which is one, i.e., primitive Hurwitz integers. We show, with the help of a proposition, that some primitive Hurwitz integers accompanied by a related modulo function are not suitable for constructing Hurwitz signal constellations. To solve this problem, we show, with the help of a proposition, the existence of primitive Hurwitz integers that have the "division with small remainder" property used to construct the Hurwitz constellations. We also call the set of these integers named as "Encoder Hurwitz Integers" set. Moreover, we examine some properties of the mentioned set. In addition, we investigate the performances of Hurwitz signal constellations, which are constructed accompanied by a related modulo function using Hurwitz integers, each component of which is in half-integers, for transmission over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Cubic Sublattices

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An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.

PERFECT CODES OVER HURWITZ INTEGERS INDUCED BY CIRCULANT GRAPHS

In this paper, a new family of t−error correcting perfect codes over Hurwitz integers is presented. To obtain these perfect codes, the perfect t−dominating sets over the circulant graphs are used. The codewords of such perfect codes are generated by the elements of a subgroup of the considered group.

Quaternionic hyperbolic lattices of minimal covolume

Abstract For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$ . We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all $n>1$ .

Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multilevel Coding

The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.

Congruence subgroups and orthogonal groups

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups SO0(2,n), n=3,4,6. The proof is based on an application of linear algebra adapted to the number theoretical needs.

Zhi-Wei Sun's 1-3-5 conjecture and variations

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhi-Wei Sun's “1-3-5 conjecture” for all integers, and reduce the general case to its verification up to 1.052×1011. The computational verification was performed by the authors and a colleague, concluding the proof of Sun's 1-3-5 conjecture. We also establish some variations of this conjecture.

Two-Stage Coded Modulation for Hurwitz Constellations in Fiber-Optical Communications

Four-dimensional signal constellations based on the checkerboard lattice $\boldsymbol{D}_4$ offer a packing gain over conventional QAM constellations per polarization. Due to the increased number of nearest neighbors such power-efficient 4D formats cannot be Gray-labeled and bit-interleaved coded modulation results in a considerable performance loss. Instead, a suited coded-modulation scheme must be tailored to the properties of the underlying signal lattice. We apply a low-complexity two-stage coded-modulation scheme for constellations based on the set of Hurwitz integers, an isomorphic representation of the $\boldsymbol{D}_4$ lattice. The proposed signaling scheme is evaluated in numerical simulations and fiber-optical system experiments. The performance is compared to a reference implementation using square QAM formats as proposed in the current 400ZR standardization activities. We demonstrate a 0.8 dB gain in required OSNR of the 512-ary Hurwitz constellation over dual-polarization 16-QAM (with 256 signal points in 4D) at the same symbol and bit rate. Additional numerical and experimental results show the potential for applications in the long-haul as well as the short-reach regime.

Generalized Multistream Spatial Modulation With Signal Constellations Based on Hurwitz Integers and Low-Complexity Detection

This letter proposes two contributions to improve the performance of transmission with generalized multistream spatial modulation (SM). In particular, a modified suboptimal detection algorithm based on the Gaussian approximation method is proposed. The proposed modifications reduce the complexity of the Gaussian approximation method and improve the performance for high signal-to-noise ratios. Furthermore, this letter introduces signal constellations based on Hurwitz integers, i.e., a 4-D lattice. Simulation results demonstrate that these signal constellations are beneficial for generalized SM with two active antennas.

Waring’s problem for integral quaternions

A (Lipschitz) integral quaternion is a Hamiltonian quaternion of the form a+bi+cj+dk with all of a,b,c,d∈Z. In 1946, Niven showed that the integral quaternions expressible as a sum of squares of integral quaternions are precisely those for which 2∣b,c,d; moreover, all of these are expressible as sums of three squares. Now let m be an integer with m>2, and suppose 2r∥m. We show that if r=0 (i.e., m is odd), then all integral quaternions are sums of mth powers, while if r>0, then the integral quaternions that are sums of mth powers are precisely those for which 2r∣b,c,d and 2r+1∣b+c+d. Moreover, all of these are expressible as a sum of g(m)mth powers, where g(m) is a positive integer depending only on m.

Volumes and distributions for random unimodular complex and quaternion lattices

Two themes associated with invariant measures on the matrix groups SLN(F), with F=R,C or H, and their corresponding lattices parametrised by SLN(F)/SLN(O), O being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of SLN(F) with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in SLN(O). The second is the problem of lattice reduction in the case N=2. A unified proof of the validity of the appropriate analogue of the Lagrange–Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With F=C this allows for the exact computation of the PDF of the first minimum (for O=Z[i] and Z[(1+−3)/2]), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for O=Z[i]). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for F=H and O the Hurwitz integers, for which direct integration was not possible.

The L-algebra of Hurwitz primes

Conway and Smith proved that up to recombination of conjugate primes and migration of units, the only obstruction to unique factorization in the ring of Hurwitz integers in the quaternions is metacommutation of primes with distinct norm. We show that the Hurwitz primes form a discrete L⁎-algebra, a quantum structure which provides a general explanation for metacommutation. L-algebras arise in the theory of Artin–Tits groups, quantum logic, and in connection with solutions of the quantum Yang–Baxter equation. It is proved that every discrete L⁎-algebra admits a natural embedding into a right ℓ-group, which yields a new class of Garside groups.

The non-orthogonal Cayley–Dickson construction and the octonionic structure of the E8-lattice

Using a conic [Formula: see text] algebra [Formula: see text] over an arbitrary commutative ring, a scalar [Formula: see text] and a linear form [Formula: see text] on [Formula: see text] as input, the non-orthogonal Cayley–Dickson construction produces a conic algebra [Formula: see text] and collapses to the standard (orthogonal) Cayley–Dickson construction for [Formula: see text]. Conditions on [Formula: see text] that are necessary and sufficient for [Formula: see text] to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of [Formula: see text] even if [Formula: see text] is singular are also given. As an application, we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley–Dickson construction.

Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

Using the rings of Lipschitz and Hurwitz integers $$\mathbb {H}(\mathbb {Z})$$ and $$\mathbb {H}ur(\mathbb {Z})$$ in the quaternion division algebra $$\mathbb {H}$$ , we define several Kleinian discrete subgroups of $$PSL(2,\mathbb {H})$$ . We define first a Kleinian subgroup $$PSL(2,\mathfrak {L})$$ of $$PSL(2,\mathbb {H}(\mathbb {Z}))$$ . This group is a generalization of the modular group $$PSL(2,\mathbb {Z})$$ . Next we define a discrete subgroup $$PSL(2,\mathfrak {H})$$ of $$PSL(2,\mathbb {H})$$ which is obtained by using Hurwitz integers. It contains as a subgroup $$PSL(2,\mathfrak {L})$$ . In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {L})$$ and $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {H})$$ which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space.

On the Zero Divisor Graphs of the Ring of Lipschitz Integers Modulo n

This article studies the zero divisor graphs of the ring of Lipschitz integers modulo n. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these graphs.

Lattice Index Coding

The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific case of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. Instances of this communication problem arise in wireless relay networks, sensor networks, and retransmissions in broadcast channels. We construct 'lattice index codes' for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a coarse lattice. We introduce a design metric called 'side information gain' that measures the advantage of a code in utilizing the side information at the receivers, and hence its goodness as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes with large side information gains using lattices over the following principal ideal domains: rational integers, Gaussian integers, Eisenstein integers, and the Hurwitz quaternions. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain. Finally, using an example, we illustrate how the proposed lattice index codes can benefit Gaussian broadcast channels with more general message demands.

New Four-Dimensional Signal Constellations From Lipschitz Integers for Transmission Over the Gaussian Channel

Codes over quotient rings of Lipschitz integers have recently attracted some attention. This work investigates the performance of Lipschitz integer constellations for transmission over the AWGN channel by means of the constellation figure of merit. A construction of sets of Lipschitz integers that leads to a better constellation figure of merit compared to ordinary Lipschitz integer constellations is presented. In particular, it is demonstrated that the concept of set partitioning can be applied to quotient rings of Lipschitz integers where the number of elements is not a prime number. It is shown that it is always possible to partition such quotient rings into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is strictly larger than in the original set. The resulting signal constellations have a better performance for transmission over an additive white Gaussian noise channel compared to Gaussian integer constellations and to ordinary Lipschitz integer constellations. In addition, we present multilevel code constructions for the new signal constellations.