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

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz 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 Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by deﬁning Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can deﬁne appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation ﬁgure of merit (CFM), average energy, and signal-to-noise ratio (SNR).


I. 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][2][3][4][5][6][7][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 greatest 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. The residual class set of the selected Gaussian integer is called a signal constellation. The "signal constellation" is a communication term. In [9], Huber is used prime Gaussian integers such that 1 ≡ p mod 4 where p = αα, and α 1 > α 2 > 0. Codes over rings of Gaussian integers were studied in papers to [9][10][11][12].
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. In [13], Ozen and Guzeltepe study codes over some finite fields by using commutative quaternion integers. The Lipschitz integers are a four-dimensional vector space that is an algebra over the set of integers. The set of Lipschitz integers is forms a subring of quaternion integers ring. A Lipschitz integer is called a primitive Lipschitz integer if the greatest common divisor(gcd) of all components is one i.e. gcd(α 1 , α 2 , α 3 , α 4 ) = 1. In [16], codes over quaternion integers, where the elements are from the commutative ring of even integers, are given. Codes over quaternion integers, and codes over Lipschitz integers were studied in papers given in [13][14][15][16][17][18][19][20]. Without loss of generality in [15], [16], we consider Lipschitz integers, which their norm is both a prime integer and not a prime integer and each component is non-zero, and such that α 1 ≥ α 2 ≥ α 3 ≥ α 4 > 0 where gcd(α 1 , α 2 , α 3 , α 4 ) = 1 in this study. In [9], Hubers' 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 algorithm, 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 norm in the residue class. Considering the original idea for a residual class of Lipschitz integer rings, any element of the signal constellation may not be represented by a Lipschitz integer with the minimal norm in *Responded Author the residue class. In other words, the norm of any element of the Lipschitz signal constellation is equal to the norm of the primitive Lipschitz integer used to obtained the Lipschitz signal constellation. In this study, we define a set that is consists of primitive Lipschitz integers that have the "division with small remainder" property to solve this problem. This set is named encoder Lipschitz integers set. We construct Lipschitz signal constellations by modulo function with encoder Lipschitz integers. Also, we investigate performances between of Lipschitz integer constellations over the additive white Gaussian noise (AWGN) channel with respect to constellation figure of merit, average energy, and signal-to-noise ratio (SNR). We close this study with the conclusion section.

II. 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 [21, 2.4.1 Definition]. Let α = α 1 + α 2 i + α 3 j + α 4 k be a quaternion. Here α 1 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 [13]. Also, multiplication of α and α −1 is commutative since The set of all Lipschitz integers that denoted by H (Z) is written by This set is a subset of H (R). H (Z) is forms a subring of the ring of all quaternions because of closed under multiplication and addition. The conjugate of The units of H (Z) under multiplication are {±1, ±i, ±j, ±k}. This set has 8 elements. We will use L notation instead of H (Z) for the rest of this study. Definition 2.3: A Lipschitz integer is called a prime Lipschitz integer just if its norm is a prime in Z. Example 2.1: α = 3 − 2i + 3j − k is a prime Lipschitz integer since N (α) = 3 2 + (−2) 2 + 3 2 + (−1) 2 = 23 is a prime in Z.
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 Lipschitz integers from a quaternions. In this study, the rounding is done in the direction +∞. Example 2.3: Let be α = 5 4 + 1 2 i − 1 2 j − 5 2 k ∈ H (R) a quaternion. If we use nearest integer rounding notation for α, then we obtain a Lipschitz integer. That is, The residue class set for codes over Gaussian integers that are two-dimensional signal space are constructed by the modulo function technique. Similar to Gaussian integers [9], Freudenberger and Shavgulidze generalized this technique for Lipschitz integers whose norm is both a prime integer and not a prime integer [14][15]. This technique, known as the modulo function, is given by the following definition. Note that we consider the primitive Lipschitz integers whose norm is both a prime integer and not a prime integer, and left residue class set of primitive Lipschitz integer in this study.
Definition 2.6: The modulo function µ : Z N (π) → L π is defined by where π is a primitive Lipschitz integer and z ∈ Z N (π) . Here Z N (π) is the well-known residual class ring of ordinary integers with N (π) elements, L π is the left residual class set of z ∈ Z N (π) modulo π, and µ π (z) is given remainder of z(z ∈ Z N (π) ) with respect to modulo π. We can also consider z as a Lipschitz integer such that its imaginary part is zero. The quotient ring of the Lipschitz integers modulo this equivalence relation, which we denote as L π = zmodπ : z ∈ Z N (π) . The L π set contains N (π) elements. If π is a prime Lipschitz integer, then the modulo function µ defines a bijective mapping from Z N (π) into L π which is a four-dimensional signal space. Therefore, the modulo function µ is a ring isomorphism between Z N (π) and L 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 Lipschitz integer, the modulo function µ is a group isomorphism with respect to addition between Z N (π) and L 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 Lipschitz integers set in the following section, we can define a ring isomorphism between Z N (π) and L π where π is an encoder Lipschitz 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 "signal constellation" term instead of "left residue class set" term. You can find more details which related to the arithmetics properties of quaternions and Lipschitz integers in [21][22].

III. ENCODER LIPSCHITZ 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 Lipschitz integers, then there exists unique Lipschitz integers β and γ such that α = πβ + γ, 0 ≤ N (γ) ≤ N (π) where α, π ∈ L. Therefore, the Euclid division algorithm does not work for Lipschitz integers because of 0 ≤ N (γ) ≤ N (π). So, the primitive Lipschitz integers do not have the "division with small remainder" property. The Euclid division algorithm works for prime Lipschitz integers [22]. Because each element in the L π has the minimal norm. So, the prime Lipschitz integers have the "division with small remainder" property. The following proposition and lemma imply that primitive Lipschitz integers do not have the "division with small remainder" property since not working Euclid division algorithm for primitive Lipschitz integers whose each component is an odd integer.
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 Lipschitz integers whose norm is not a prime integer with the following lemma.

Lemma 3.2:
Let π be a primitive Lipschitz integer whose norm is not a prime integer. Then, where α is a Lipschitz integer.
Example 3.1: Let π = 3+i+5j+3k be a Lipschitz integer. Also, π is a primitive Lipschitz integer because of gcd(3, 1, 5, 3) = 1. By equation (3), the Lipschitz constellation is This set contains 44 elements because of N (π) = 3 2 + 1 2 The norm of other elements in the set (14) is less than the norm of π. So, primitive Lipschitz integer π = 3 + i + 5j + 3k does not have the "division with small remainder" property. In addition, to be a Euclidean metric, But this is not true. Consequently, the Euclidean metric is not provide for the L π constellation constructed by π = 3+i+5j +3k Lipschitz integer, and the Euclidean division algorithm is not work for π = 3 + i + 5j + 3k Lipschitz integer.
We define a set that consists of the primitive Lipschitz integers that have the "division with small remainder" property with the following definition. This set is a subset of the primitive Lipschitz integers, and Lipschitz integers.
The following theorem demonstrates that the encoder Lipschitz integers have the "division with small remainder" property. Proposition 3.5: Let π is an encoder Lipschitz integer. Then, Proof 3.5: We shall analyze and prove this theorem case by case. Let π = π 1 + π 2 i + π 3 j + π 4 k is an encoder Lipschitz integer.
This completes this proof. With the following examples, given an example for each case in the proposition 3.5. Example 3.2: Case 1 π = 1 + 3i + 2j + k is an encoder Lipschitz integer. By equation (3), the Lipschitz integer constellation is µπ (4) = −1 + 2j + k, µπ (5) = 2j + k, µπ (6) = 1 + 2j + k, µπ (7) = 2 + 2j + k, µπ (8) The set contains 15 elements because of N (π) = 1 2 + 3 2 + 2 2 + 1 2 = 15. The norm of each element in the set is less than the norm of π. Example 3.3: Case 2 π = 2 + 3i + j + 2k is an encoder Lipschitz integer. By equation (3), the Lipschitz integer constellation is µπ (10) = −1 + 2j + k, µπ (11) = 2j + k, µπ (12) = 1 + 2j + k, µπ (13) = 2 + 2j + k, µπ (14) The set contains 18 elements because of N (π) = 2 2 + 3 2 + 1 2 + 2 2 = 18. The norm of each element in the set is less than the norm of π. Example 3.4: Case 3 π = 2+3i+2j +2k is an encoder Lipschitz integer. By equation (3), the Lipschitz integer constellation is The set contains 21 elements because of N (π) = 2 2 + 3 2 + 2 2 + 2 2 = 21. The norm of each element in the set is less than the norm of π. 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 Lipschitz integers in example 3.2, example 3.3, and example 3.4. As a result of these examples, we represent the following proposition. Proposition 3.6: Let π be an encoder Lipschitz integer. Then, where α is a Lipschitz integer, and µ π (z) + µ π (N (π) − z) = 0 where z ∈ Z N (π) . Proof 3.6: Firstly, let show us that µ π (α) + µ π (N (π) − α) = 0 where π is an encoder Lipschitz integer, and α is a Lipschitz integer. By equation (3), we have ⌊π − απ N (π) ⌉ = ⌊π⌉ − ⌊ απ N (π) ⌉ = π − ⌊ απ N (π) ⌉ since π is an encoder Lipschitz integer. Consequently, Now, let show us that µ π (z) + µ π (N (π) − z) = 0 where π is an encoder Lipschitz integer and, z ∈ Z N (π) . By equation (3), we have IV. PERFORMANCES OF LIPSCHITZ CONSTELLATIONS FOR TRANSMISSION OVER AWGN CHANNEL In this section, we are first giving some distance and performance measures, and then we investigate the performance of Lipschitz constellations that lies on encoder Lipschitz integers for transmission over AWGN channel by agency of average energy, CFM, and SNR gains. Also, we give set partitioning property on larger Lipschitz integers,namely proposed Lipschitz integers. You can found more further information in [14][15]. We follow the procedures in [15] for some distance, performance measures and set partitioning property.. The average energy of a constellation denoted by E π is computed by The squared Euclidean distance of two Lipschitz integers is defined as and the minimum squared Euclidean distance of the constellation is where α, β ∈ L π . In [23], 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 twodimensions. So, the CFM of a M −dimensional constellation is computed by A higher CFM leads to a better performance for transmission over an AWGN channel. 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).
The SNR gains of a Lipschitz constellation over the AWGN channel is computed by where Lipschitz constellation L π and Gaussian constellation are the equal size. A residue class ring of Lipschitz integers L 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 Lipschitz constellation. We can partition the set L π into c subsets L where Lipschitz signal constellation L (0) π and the Lipschitz constellation are the equal size. In [15], [24], Freudenberger et al. investigated Lipschitz signal constellations according to their performance for transmission over the AWGN channel by the agency of CFM. In [24], Freudenberger et al. presented a comparison of all Gaussian and Lipschitz signal constellations of the same size for N (π) < 100, where both signal constellations exist. They demonstrated that both signal constellations have almost same CFM. Thereby they concluded that these Lipschitz signal constellations have no performance advantage for transmission over an AWGN channel. To Lipschitz signal constellations have performance advantage for transmission over an AWGN channel, they constructed new Lipschitz signal constellations based on set partitioning of larger signal sets. In [15], Freudenberger et al. investigated proposed Lipschitz signal constellations that have set partitioning property according to their performance for transmission over the AWGN channel by the agency of CFM. We present different proposed primitive Lipschitz integers that have the "division with small remainder" in Table I instead of the proposed primitive Lipschitz integers that do not have the "division with small remainder" property used to constructed proposed Lipschitz constellations in [15, Table I]. You can examine the below example 4.1 and example 4.2.
Example 4.1: In [15, Table I], Freudenberger et al. presented 9 + 7i + 5j + k proposed primitive Lipschitz integer with N = c · d = 6 · 26 = 156. They gave CFM and SNR gain of proposed Lipschitz constellation that has 156 elements with respect to primitive Lipschitz constellation with 26 elements, which is a subset that has a larger minimum square Euclidean distance of proposed Lipschitz constellation with 156 elements. But proposed primitive Lipschitz integer with 156 elements does not has the "division with small remainder" property since the Euclid division algorithm does not work with respect to proposition 3.1. Also, all proposed primitive Lipschitz integers with N = c · d = 6 · 26 = 156 do not have the "division with small remainder" property. These proposed primitive Lipschitz integers with N = c · d = 6 · 26 = 156 are 7 + 7i + 7j + 3k, 9 + 5i + 5j + 5k, 9 + 7i + 5j + k, and 11 + 5i + 3j + k. So, instead of 9 + 7i + 5j + k proposed primitive Lipschitz integer, we propose 10 + 5i + 2j + k proposed primitive Lipschitz integer with N = c · d = 5 · 26 = 130. This proposed primitive Lipschitz integer has the "division with small remainder" property since it will be verified by proposition 3.5. Shortly, we propose 10 + 5i + 2j + k encoder Lipschitz integer in Table I instead of the proposed primitive Lipschitz integer that is given for d = 26 since the proposed primitive Lipschitz integer that is given for d = 26 in [15, Table I] does not has the "division with small remainder" property. We present CFM and SNR gain of this proposed Lipschitz constellation with respect to primitive Lipschitz constellation with 26 elements, which is a subset that has a larger minimum square Euclidean distance of this proposed Lipschitz constellation in Table I proposed primitive Lipschitz integer with N = c · d = 10 · 74 = 740. They gave CFM and SNR gain of this proposed Lipschitz constellation with respect to primitive Lipschitz constellation with 74 elements, which is a subset that has a larger minimum square Euclidean distance of this proposed Lipschitz constellation. But this proposed primitive Lipschitz integer does not has the "division with small remainder" property since the Euclid division algorithm does not work with respect to proposition 3.1. Also, all proposed primitive Lipschitz integers with N = c · d = 10 · 74 = 740 do not have the "division with small remainder" property. These proposed primitive Lipschitz integers with N = c · d = 10 · 74 = 740 are 15 + 15i + 13j + 11k, 17 + 15i + 15j + k, 17 + 17i + 9j + 9k, 19 + 19i + 3j + 3k, 21 + 13i + 9j + 7k, 21 + 13i + 11j + 3k, 21 + 15i + 7j + 5k, 21 + 17i + 3j + k, 23 + 9i + 9j + 7k, 23 + 11i + 9j3k, 25 + 9i + 5j + 3k and 27 + 3i + j + k. So, instead of 19 + 17i + 9j + 3k proposed primitive Lipschitz integer, we propose 17 + 14i + 10j + 9k proposed primitive Lipschitz integer with N = c · d = 9 · 74 = 666. This proposed primitive Lipschitz integer has the "division with small remainder" property since it will be verified by proposition 3.5. The Euclid division algorithm works for this encoder Lipschitz integer. We present CFM and SNR gain of this proposed Lipschitz constellation with respect to primitive Lipschitz constellation with 26 elements, which is a subset that has a larger minimum square Euclidean distance of this proposed Lipschitz constellation in Table I. You can see Table II for other proposed primitive Lipschitz integers with N = c · d = 9 · 74 = 666. Example 4.3: In Table II, we present other proposed primitive Lipschitz integers used to construct proposed Lipschitz constellations that have higher CFM for some N values that have set partitioning property. These proposed primitive Lipschitz integers have the "division with small remainder" property since it will be verified by proposition 3.5. We do not present proposed Lipschitz integers used to construct proposed primitive Lipschitz constellations with d = 10, 13, 25, 26, 34, 65, or 89 elements in this paper since there exists just one. You can see [15, Table II and Table III. In Table III, we present all primitive Lipschitz integers used to construct Lipschitz constellations with d < 100 elements, and CFM, average energy, and SNR gain of Lipschitz constellations that have same size with Gaussian constellations.
The below examples are given clues about the construction of tables. Example 4.5: We consider proposed Lipschitz constellation(s) with N = 3 · 13 = 39 elements. There exist two different proposed primitive Lipschitz integers used to construct proposed Lipschitz constellations with N = 39. These proposed primitive Lipschitz integers are 5 + 3i + 2j + k and 6 + i + j + k. There is no Gaussian constellation that has an equal size with the proposed Lipschitz constellation that has N = 39 elements. Note that proposed primitive Lipschitz integers are not to be the same size as primitive Gaussian integers to apply set partitioning property for proposed primitive Lipschitz integers. For L 5+3i+2j+k proposed Lipschitz constellation, the minimum squared Euclidean distance, CFM and average energy are 1, 12.9744 and 0.1542, respectively. The L 5+3i+2j+k proposed Lipschitz constellation is partition the c = 3 different subsets with each set d = 13 elements. So, the SNR coding gain of L 5+3i+2j+k proposed Lipschitz constellation is computed by using a subset of L 5+3i+2j+k proposed Lipschitz constellation. This subset is the set L Lipschitz signal constellation that is a subset of L 5+3i+2j+k Lipschitz constellation are 9, 12.9231 and 1.3929, respectively. Also, the proposed primitive Lipschitz integer used to construct the Lipschitz constellation that is the equivalence L (0) 5+3i+2j+k constellation with 13 elements is 2 + 2i + 2j + k (see Table III). The minimum squared Euclidean distance, CFM and average energy of L 2+2i+2j+k Lipschitz constellation is 1, 0.4643, and 4.3077. The minimum squared Euclidean distance, CFM and average energy of the Gaussian constellation G 3+2i with 13 elements are 1, 2.1539 and 0.4643, respectively. Therefore, SNR gain of L 5+3i+2j+k proposed Lipschitz constellation is SNR = −10 log( CFM of L  CFM and the minimum squared Euclidean distance. So, the proposed primitive Lipschitz integer used to construct proposed Lipschitz constellation that has higher CFM and larger minimum squared Euclidean distance between subsets of L 5+3i+2j+k and L 6+i+j+k proposed Lipschitz constellations is 5 + 3i + 2j + k. For transmission over AWGN channel, the L 5+3i+2j+k proposed Lipschitz signal constellation has better performance than the L 6+i+j+k proposed Lipschitz signal constellation.

V. CONCLUSION
In this study, we investigated Lipschitz integers that have "division with small remainder" property and defined a new set, which is formed Lipschitz integers that have "division with small remainder" property, named encoder Lipschitz integers. So, we can define a Euclidean metric for Lipschitz constellations that lies on the Lipschitz integers or, different appropriate metrics for codes over the rings of Lipschitz integers. We showed that the Euclid division algorithm whether works or not for Lipschitz integers with proposition 3.1 and proposition 3.5. You can check whether the Euclidean division algorithm works for Lipschitz integers with these propositions (see example 4.1 and example 4.2). Also, we examined the performances of the Lipschitz integers, which have the "division with small remainder" property, for transmission over the AWGN channel, and proposed new Lipschitz constellations for transmission over AWGN channel (see Table I).