Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

. The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by deﬁning Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can deﬁne appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, 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).


Introduction
In recent years, many researchers in coding theory have investigated some special code constructions over groups, fields or rings i.e. finite algebraic construction. Specially, they have studied code constructions over finite rings of integers and finite fields [1]- [8]. A Gaussian integer is a complex number that real and imaginary parts are in Z. The set of Gaussian integers that denoted by Z[i] is shown by Z[i] = {α = α 1 + α 2 i : α 1 , α 2 ∈ Z, i 2 = −1}. Gaussian integers are a commutative ring and a subset of the complex numbers field, since they are closed under addition and multiplication. Let α = α 1 + α 2 i be a Gaussian integer. The conjugate of α is equal to α = α 1 − α 2 i, the norm of α is equal to N (α) = α 2 1 + α 2 2 ,, and the inverse of α is equal to α −1 = α N (α) where its norm is non-zero. A Gaussian integer is a prime Gaussian integer if its norm is a prime in Z. A Gaussian integer is a primitive Gaussian integer just if greater common divisor (gcd) of all components is one i.e. gcd(α 1 , α 2 ) = 1. Hence, α 1 and α 2 are positive integers if α = α 1 + α 2 i is a primitive Gaussian integer. In [9], first study about code constructions over Gaussian integers is presented by Huber. In other words, Huber discovered a new way to construct codes for two dimensional signals by virtue of Gaussian integers, i.e. the integral points on the complex plane [9]. His original idea is to regard a finite field as a residue class of the Gaussian integer ring modulo a prime Gaussian integer and, by Euclidean division, to get a unique element of minimal norm in each residue class, which represents each element of a finite field. Therefore, each element of a finite field can be represented by a Gaussian integer with the minimal Galois norm in the residue class, and the residual class set of the selected Gaussian integer is called a signal constellation. The "signal constellation" is a communication term. Huber is used prime Gaussian integers such that 1 ≡ p mod 4 where p = αα, and α 1 > α 2 > 0. In this study, we use primitive Gaussian integers that are both a prime integer and such that α 1 > α 2 > 0. Codes over rings of Gaussian integers were studied in papers to [9]- [13].
The quaternions are a four dimensional vector space that is an algebra over the set of the real numbers (R) , and a number system that extends the complex numbers (C). The quaternions are a division algebra that is associative and non-commutative since the multiplication of quaternions has not commutative property. So, αβ = βα where α and β are quaternions. Let α = α 1 + α 2 i + α 3 j + α 4 k be a quaternion where α 1 is real part and α 2 i + α 3 j + α 4 k is imaginary part. Multiplication of quaternions has commutative properties when αα −1 = α −1 α = 1, and their imaginary parts are parallel to each other. The coding techniques in [9] have been generalized to codes over quaternion integers. In [14], Ozen and Guzeltepe study codes over some finite fields by using commutative quaternion integers. Codes over rings of quaternion integers were studied in papers to [13]- [17]. In this study, we consider Hurwitz integers, which are four dimensional signal constellations that are quotient rings. α 1 + α 2 i + α 3 j + α 4 k is a Hurwitz integer just if either all of α 1 , α 2 , α 3 , and α 4 are in Z or all in Z + 1 2 . In [18], Guzeltepe studied classes of linear codes over Hurwitz integers equipped with a new metric that refer as the Hurwitz metric. In [19], Rohweder et al. presented a new algebraic construction of finite sets of Hurwitz integers by a respective modulo function, and investigated performance for transmission over the additive white Gaussian noise (AWGN) channel,which is a noise channel model. The codes over Hurwitz integers given in [18][19][20][21][22].
This work is organized as follows: In the next section, we give some fundamental information about quaternions and Hurwitz integers. Also, we give the modulo function used to establish the notation and the notion of a residual class of Hurwitz integers ring with respect to primitive Hurwitz integers. In Section III, we define a set that is consists of primitive Hurwitz integers that have the "division with small remainder" property. This set is named encoder Hurwitz integers set. In Section IV, we investigate the performance of Hurwitz constellations for transmission over the additive white Gaussian noise by means of constellation figure of merit (CFM), average energy, and signal-noise-to ratio (SNR). Finally, we conclude the paper in Section V.

Preliminaries
In this section, we give some fundamental information used throughout this paper.
Definition 2.1. The Hamilton quaternion algebra over R, is the associative unital algebra given by the following representation: i. H (R) is the free R− module over the symbols 1, i, j, k, that is: The definition is natural, in the sense that any unital ring homomorphism R 1 → R 2 extends to a unital ring homomorphism H(R 1 ) → H(R 2 ) by mapping 1 to 1, i to i, j to j and k to k [23, 2.5.1 Definition]. Let α = α 1 +α 2 i+α 3 j+α 4 k be a quaternion. Here α 0 is a real part ,and α 2 i+α 3 j+α 4 k is an imaginary part. Also, α 1 , α 2 , α 2 , and α 2 are components of α quaternion. Multiplication of quaternions is non-commutative. But, if the imaginary parts of quaternions are parallel to each other, then multiplication of quaternions is commutative [14]. Also, multiplication of α and α −1 is commutative since αα −1 = α −1 α = 1.
The set of all Hurwitz integers that denoted by H (Z) is shown by The ring of Hurwitz integers H(Z) is forms a subring of the ring of all quaternions because of closed under multiplication and addition. Let α = α 1 +α 2 i+ α 3 j +α 4 k is a Hurwitz integer. The conjugate of α is α = α 1 −α 2 i−α 3 j −α 4 k, the norm of α is N (α) = α · α = α 2 1 + α 2 2 + α 2 3 + α 2 4 , and the inverse of α is . So α is a unit of H such that N (α) = 1. The set has 24 elements. Definition 2.3. α = α 1 + α 2 i + α 3 j + α 4 k a prime Hurwitz integer just if its norm is a rational prime integer.
Definition 2.5. The nearest integer rounding notation denoted by ⌊·⌉ is defined as rounding a rational number to the integer closest to its.For quaternions, each component of a quaternion is separately rounding to the integer closest to its. So, we obtain Hurwitz integers whose components are in Z from a quaternions. Note that the rounding is done in the direction +∞ in this study.
. If we use nearest integer rounding notation for α, then we obtain a Hurwitz integer whose components are in Z. That is, The residue class set for codes over Gaussian integers that are twodimensional signal space are constructed by the modulo function technique. Similarity Lipschitz constellation for codes over Lipschitz integers [25], we use this technique to construct Hurwitz constellation that lies on Hurwitz integers. This technique, known as the modulo function, is given by the following definition. Note that we consider the primitive Hurwitz integers whose norm is both a prime integer and not a prime integer, and left residue class set of primitive Hurwitz integer in this study.
Definition 2.6. The modulo function µ : Z N (π) → H π is defined by where π is a primitive Hurwitz integer and z ∈ Z N (π) . Here Z N (π) is the well-known residual class ring of ordinary integers with N (π) elements, H π is the left residual class set of z modulo π, and µ π (z) is given remainder of z with respect to modulo π. We can also consider z as a Hurwitz integer such that its imaginary part is zero where z ∈ Z. The quotient ring of the Hurwitz integers modulo this equivalence relation, which we denote as H π = zmodπ : z ∈ Z N (π) . The H π set contains N (π) elements. If π is a prime Hurwitz integer, then the modulo function µ defines a bijective mapping from Z N (π) into H π which is a four-dimensional signal space. Therefore, the modulo function µ is a ring isomorphism between Z N (π) and H π . Because there exists a inverse map [8] and we have µ(z 1 + z 2 ) = µ(z 1 ) + µ(z 2 ) and µ(z 1 z 2 ) = µ(z 1 )µ(z 2 ) for any z 1 , z 2 ∈ Z N (π) . If π is a primitive Hurwitz integer, the modulo function µ is a group isomorphism with respect to addition between Z N (π) and H π . Because there exists a inverse map [8] and we have µ(z 1 +z 2 ) = µ(z 1 )+µ(z 2 ) for any z 1 , z 2 ∈ Z N (π) . After we define encoder Hurwitz integers set in the following section, we can define a ring isomorphism between Z N (π) and H π where π is an encoder Hurwitz integer.
In engineering, the "signal constellation" has been used as a communication term. In mathematics, the "signal constellation" means for residue class set. In the rest of this study, we use the" signal constellation" term instead of the" left residue class set" term. You can find more details which related to the arithmetics properties about arithmetic properties of quaternions and Hurwitz integers in [23][24].

Encoder Hurwitz Integers
The Euclid division algorithm says that there exists unique integers q and r such that a = bq + r, 0 ≤| r |<| b | where a, b ∈ Z. Here a is the dividend, b is the divisor, q is the quotient, r is the remainder, and | · | is the symbol for absolute value. If we generalize the Euclid division algorithm for Hurwitz integers whose components are in Z, then there exists unique Hurwitz integers β and γ such that α = πβ + γ, 0 ≤ N (γ) ≤ N (π) where α, π ∈ H such that their components are in Z. Therefore, the Euclid division algorithm does not work for Hurwitz integers whose components are in Z because of 0 ≤ N (γ) ≤ N (π). So, the primitive Hurwitz integers whose components are in Z do not have the "division with small remainder" property. If we generalize the Euclid division algorithm for Hurwitz integers whose components are in Z + 1 2 , then there exists unique Hurwitz integers β and γ such that α = πβ +γ, 0 ≤ N (γ) < N (π) where α, π ∈ H such that their components are in Z + 1 2 . Therefore, the Euclid division algorithm works for Hurwitz integers whose components are in Z + 1 2 because of 0 ≤ N (γ) < N (π). So, the primitive Hurwitz integers whose components are in Z + 1 2 have the "division with small remainder" property [24]. Also, the Euclid division algorithm generally works for prime Hurwitz integers. Because each element in the H π has the minimal norm. So, the prime Hurwitz integers have the "division with small remainder" property. The following proposition and lemma imply that primitive Hurwitz integers whose components are in Z do not have the "division with small remainder" property since not working Euclid division algorithm for primitive Hurwitz integers whose each component is an odd integer.
Proposition 3.1. Let π is a primitive Hurwitz integer whose each component is an odd integer. Then, with respect to equation (2.3).
Let γ = µ π ( N (π) 2 ). By proposition 3.1, we have N (µ π ( N (π) 2 )) = N (γ) = N (π). In other words, the norm of the remainder is equal to the norm of the divisor. We generalize proposition 3.1 for primitive Hurwitz integers whose norm is not a prime integer where its each component is in Z with the following lemma.
Lemma 3.1. Let π be a primitive Hurwitz integer, and α be a Hurwitz integer. If π is a primitive Hurwitz integer whose all component are in Z, then with respect to equation (2.3).
Proof. Let π be a primitive Hurwitz integer, and α = α 1 + α 2 i + α 3 j + α 4 k be a Hurwitz integers which its norm is non-zero. If π is a primitive Hurwitz integer whose all component are in Z, by equation (2.3) Therefore, we have Hereby, (3.10) This completes the proof.
Consequently, primitive Hurwitz integers, whose norm is not a prime integer, such that each component is in Z, do not have the "division with small remainder" property. So, the Euclid division algorithm does not work for primitive Hurwitz integers, whose norm is not a prime integer, such that each component is in Z. With the following example, given practices for proposition 3.1.
We define a set that consists of the primitive Hurwitz integers that have the "division with small remainder" property with the following definition. This set is a subset of the primitive Hurwitz integers, and Hurwitz integers.
The definition 3.1 is a flexible definition. Namely, the elements of the encoder Hurwitz integers set are expandable or collapsible depending on the used modulo technique. In this study, we defined the above definition with respect to the modulo function defined in definition 2.6. Let now us show that the modulo function µ defined between Z N (π) and H π by equation (2.3) is a ring isomorphism with the following theorems.
With the following examples, giving an example for each case in the proposition 3.2.
Example 3.2, example 3.3, and example 3.4 are verified all the conditions for it to be an Euclidean metric. Also the Euclidean division algorithm works for the Hurwitz integers in example 3.2, example 3.3, and example 3.4. As a result of these examples, we represent the following proposition.
In the following example, we show that it does not have the "division with small remainder" property of a Hurwitz integer used to obtain the Hurwitz constellation constructed with a different technique by Rohweder et al..
The technique in [19] is more appropriate for the Lipschitz integers whose norm is an odd number but inappropriate for the Lipschitz integers whose norm is an even number. Note that the definition 3.1 is a flexible definition. According to the modulo technique in [19], we can define the set of the encoder Hurwitz integers with "The Hurwitz integer whose the greatest common divisor of its components is one and its norm is an odd number is called an encoder Hurwitz integer.". The set of encoder Hurwitz integers is the set of the Hurwitz integers remaining with taking out the Hurwitz integers providing proposition 3.1 from the Hurwitz integers set, in general. We refer to proposition 3.1 and example 3.5 for this general definition. Note that we consider the definition 3.1 in this study. Note that we consider definition 3.1 in this study. We can come to a conclusion that the Euclid division algorithm works for the elements of the encoder Hurwitz integers set. So, we can construct well-defined Hurwitz constellations in terms of algebraic constructions for codes over Hurwitz integers. Consequently, we should use proposition 3.1 to check whether the Hurwitz integers used to construct Hurwitz constellations have the "division with small remainder" property or not. Consequently, we should use proposition 3.1 to check whether the Hurwitz integers used to construct Hurwitz constellations have the "division with small remainder" property or not.

Performances of Hurwitz Constellations for Transmission over AWGN channel
In this section, we are first giving some distance and performance measures, and then we investigate the performances of Hurwitz constellations that lies on encoder Hurwitz integers for transmission over AWGN channel by agency of average energy, CFM, and SNR gains. Note that we investigate the performances of Hurwitz constellations constructed with Hurwitz integers whose components are in Z + 1 2 by using the technique used for Gaussian constellations in [9] and Lipschitz integers in [25] in this study. Because the Hurwitz constellations constructed of primitive Hurwitz integers which their components are in Z show the same performances with Lipschitz constellations in [25]. Therefore, we give set partitioning property on larger Hurwitz integers namely, proposed Hurwitz integers, since the Hurwitz constellations that have the same size with Gaussian constellations are almost shown the same performances for transmission over the AWGN channel. We follow the procedures in [25] for some distance, performance measures and set partitioning property. The average energy of a constellation denoted by E π is computed by (4.1) The squared Euclidean distance of two Hurwitz integers is defined as and the minimum squared Euclidean distance of the constellation is where α, β ∈ H π . In [26], Forney and Wei proposed the constellation figure of merit (CFM) to compare signal constellations of different dimensions. The CFM is the ratio of the minimum squared Euclidean distance and the average energy per two-dimensions. So, the CFM of a M −dimensional constellation is computed by (4.4) A higher CFM leads to a better performance for transmission over an AWGN channel [24]. Asymptotic coding gain means for higher signal to noise ratio (SNR) [9]. The SNR of M −dimensional constellation is computed by SN R = −10 · log 10 (CFM of signal constellation). (4.5) The SNR gains of a Hurwitz constellation over the AWGN channel is SN R = −10 · log 10 ( CFM of Hurwitz signal constellation CFM of Gaussian noise constellation ). (4.6) Note that the number of elements of the Hurwitz constellation and Gaussian constellation should be the same to compare performances over the AWGN channel. A residue class ring of Hurwitz integers H π arises from the residue class ring of integers Z N (π) = {0, 1, . . . , N (π) − 1} for an integer N (π). If N (π) is not a prime integer, then we can partition the set Z N (π) into subsets of equal size. Let N = c · d where N is the elements number of Hurwitz constellation. We can partition the set H π into c subsets H Note that the number of elements of the Hurwitz constellation and Gaussian constellation should be the equal size to compare performances over the AWGN channel but proposed Hurwitz constellations should not be. Hence, we can apply set partitioning property on proposed primitive Hurwitz constellations. The SNR gains of a proposed Hurwitz constellation over the AWGN channel is computed by SN R = −10 · log 10 ( where Hurwitz signal constellation H (0) π and the Hurwitz constellation are the equal size. Guzeltepe [18], and Rohweder et al. [19] separately presented different techniques for Hurwitz constellations. Guzeltepe [18] investigated performances of the Hurwitz constellations with N (π) 2 elements where π is a primitive Hurwitz integer, over the AWGN channel by using isomorphism between H π 2 and Z π 2 . You can see example 3.5, or [19] for the technique of Rohweder et al.. Note that we use isomorphism between H π and Z π in this study.
Example 4.1. In Table I, we present the performance of the Hurwitz constellation constructed by encoder Hurwitz integers whose each component is in Z + 1 2 over the AWGN channel by means of average energy, CFM, and SNR coding gains. In Table I, the Hurwitz constellations obtained from the modulo function technique in this study have almost similar properties as Lipschitz constellations in the paper of Freudenberger et al. in [25]. The performance of Hurwitz constellations over the AWGN channel in Table I is not so good but better than nothing with respect to the performance of the Hurwitz constellations whose components is in Z over the AWGN channel. You can see [26] for the performance of the Hurwitz constellations whose components is in Z over the AWGN channel. Because the performances of Hurwitz constellations whose components is in Z and the performances of Lipschitz constellations are the same. Similarity, the performances of proposed Hurwitz constellations whose components is in Z are the same with the performances of proposed Lipschitz constellations in [25, Table I].  Table II, we present the performance of the proposed Hurwitz constellation constructed by encoder Hurwitz integers whose each component is in Z + 1 2 over the AWGN channel by means of average energy, CFM, and SNR coding gains. The proposed Hurwitz constellations in Table  II have 13 2 + 11 2 i + 7 2 j + 3 2 k proposed primitive Hurwitz integer to represent in Table I, and 9 2 + 5 2 i + 3 2 j + 1 2 k primitive Hurwitz integer to represent in Table II to create regular tables that are not crowded.

Conclusion
In this study, we investigated Hurwitz integers that have "division with small remainder" property and defined a new set, which is formed Hurwitz integers that have "division with small remainder" property, named encoder Hurwitz integers. So, we can define a Euclidean metric for Hurwitz constellations that lies on the Hurwitz integers or, different appropriate metrics for codes over the rings of Hurwitz integers. We showed that the Euclid division algorithm whether works or not for Hurwitz integers whose coefficients are in Z with proposition 3.1 and proposition 3.2. Whichever technique we use, we can check whether the Euclidean division algorithm works for Hurwitz integers with these propositions (see example 3.5). In addition, we examined the performances of the Hurwitz integers whose components are in Z + 1 2 of transmission over the AWGN channel. Also, the modulo function defined in this paper shows an inappropriate technique for Hurwitz constellations. New techniques can improvement such as Rohweder et al. [19], and Guzeltepe [18]. In our forward study, we will be investigated the performances of Hurwitz constellations for transmission over the AWGN channel by a new technique such as technique in [18]. Therefore, this paper is written to be a reference in our following study and colleagues' forward studies.