Encoder Hurwitz Integers: The Hurwitz integers that have ...

Encoder Hurwitz Integers: The Hurwitz integers thathave the ”division with small remainder” propertyRamazan Duran ( [email protected] )

Department of Mathematics, Afyon Kocatepe University https://orcid.org/0000-0002-8076-0557Murat Güzeltepe

Department of Mathematics, Sakarya University

Research Article

Keywords: Quaternion integer, Hurwitz integer, residue class, signal constellation, code construction

Posted Date: December 20th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-1053551/v4

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

https://doi.org/10.21203/rs.3.rs-1053551/v4

mailto:[email protected]

https://orcid.org/0000-0002-8076-0557

https://doi.org/10.21203/rs.3.rs-1053551/v4

https://creativecommons.org/licenses/by/4.0/

Encoder Hurwitz Integers: The Hurwitz in-

tegers that have the ”division with small re-

mainder” property

Ramazan Duran and Murat Guzeltepe

Abstract. The residue class set of a Hurwitz integer is constructed bymodulo function with primitive Hurwitz integer whose norm is a primeinteger, i.e. prime Hurwitz integer. In this study, we consider primitiveHurwitz integer whose norm is both a prime integer and not a primeinteger. If the norm of each element of the residue class set of a Hurwitzinteger is less than the norm of the primitive Hurwitz integer used toconstruct the residue class set of the Hurwitz integer, then, the Eucliddivision algorithm works for this primitive Hurwitz integer. The Eucliddivision algorithm always works for prime Hurwitz integers. In otherwords, the prime Hurwitz integers and halves-integer primitive Hurwitzintegers have the ”division with small remainder” property. However,this property is ignored in some studies that have a constructed Hur-witz residue class set that lies on primitive Hurwitz integers that theirnorms are not a prime integer and their components are in integers set.In this study, we solve this problem by defining Hurwitz integers thathave the ”division with small remainder” property, namely, encoder Hur-witz integers set. Therefore, we can define appropriate metrics for codesover Lipschitz integers. Especially, Euclidean metric. Also, we investi-gate the performances of Hurwitz signal constellations (the left residueclass set) obtained by modulo function with Hurwitz integers, whichhave the ”division with small remainder” property, over the additivewhite Gaussian noise (AWGN) channel by means of the constellationfigure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Keywords. Quaternion integer, Hurwitz integer, residue class, signal con-stellation, code construction.

1. Introduction

In recent years, many researchers in coding theory have investigated somespecial code constructions over groups, fields or rings i.e. finite algebraic

2 Ramazan Duran and Murat Guzeltepe

construction. Specially, they have studied code constructions over finite ringsof integers and finite fields [1]-[8]. A Gaussian integer is a complex numberthat real and imaginary parts are in Z. The set of Gaussian integers thatdenoted by Z[i] is shown by Z[i] = {α = α1 + α2i : α1, α2 ∈ Z, i2 =−1}. Gaussian integers are a commutative ring and a subset of the complexnumbers field, since they are closed under addition and multiplication. Letα = α1 + α2i be a Gaussian integer. The conjugate of α is equal to α =α1 − α2i, the norm of α is equal to N(α) = α2

1 + α22,, and the inverse of

α is equal to α−1 = αN(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 integeris a primitive Gaussian integer just if greater common divisor (gcd) of allcomponents is one i.e. gcd(α1, α2) = 1. Hence, α1 and α2 are positive integersif α = α1 +α2i is a primitive Gaussian integer. In [9], first study about codeconstructions over Gaussian integers is presented by Huber. In other words,Huber discovered a new way to construct codes for two dimensional signals byvirtue 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 Gaussianinteger ring modulo a prime Gaussian integer and, by Euclidean division, toget a unique element of minimal norm in each residue class, which representseach element of a finite field. Therefore, each element of a finite field canbe represented by a Gaussian integer with the minimal Galois norm in theresidue class, and the residual class set of the selected Gaussian integer iscalled a signal constellation. The ”signal constellation” is a communicationterm. Huber is used prime Gaussian integers such that 1 ≡ p mod 4 wherep = αα, and α1 > α2 > 0. In this study, we use primitive Gaussian integersthat are both a prime integer and such that α1 > α2 > 0. Codes over ringsof Gaussian integers were studied in papers to [9]-[13].

The quaternions are a four dimensional vector space that is an algebraover the set of the real numbers (R) , and a number system that extendsthe complex numbers (C). The quaternions are a division algebra that isassociative and non-commutative since the multiplication of quaternions hasnot commutative property. So, αβ 6= βα where α and β are quaternions. Letα = α1 + α2i + α3j + α4k be a quaternion where α1 is real part and α2i +α3j + α4k is imaginary part. Multiplication of quaternions has commutativeproperties when αα−1 = α−1α = 1, and their imaginary parts are parallelto each other. The coding techniques in [9] have been generalized to codesover quaternion integers. In [14], Ozen and Guzeltepe study codes over somefinite fields by using commutative quaternion integers. Codes over rings ofquaternion integers were studied in papers to [13]-[17]. In this study, weconsider Hurwitz integers, which are four dimensional signal constellationsthat are quotient rings. α1+α2i+α3j+α4k is a Hurwitz integer just if eitherall of α1, α2, α3, and α4 are in Z or all in Z + 1

2 . In [18], Guzeltepe studiedclasses of linear codes over Hurwitz integers equipped with a new metricthat refer as the Hurwitz metric. In [19], Rohweder et al. presented a newalgebraic construction of finite sets of Hurwitz integers by a respective modulo

Encoder Hurwitz Integers 3

function, and investigated performance for transmission over the additivewhite Gaussian noise (AWGN) channel,which is a noise channel model. Thecodes over Hurwitz integers given in [18-22].

This work is organized as follows: In the next section, we give some fun-damental information about quaternions and Hurwitz integers. Also, we givethe modulo function used to establish the notation and the notion of a resid-ual class of Hurwitz integers ring with respect to primitive Hurwitz integers.In Section III, we define a set that is consists of primitive Hurwitz integersthat have the ”division with small remainder” property. This set is namedencoder Hurwitz integers set. In Section IV, we investigate the performanceof Hurwitz constellations for transmission over the additive white Gaussiannoise by means of constellation figure of merit (CFM), average energy, andsignal-noise-to ratio (SNR). Finally, we conclude the paper in Section V.

2. Preliminaries

In this section, we give some fundamental information used throughout thispaper.

Definition 2.1. The Hamilton quaternion algebra over R, is the associativeunital algebra given by the following representation:

i. H (R) is the free R− module over the symbols 1, i, j, k, that is: H (R) ={a0 + a1i+ a2j + a3k : a0, a1, a2, a3 ∈ R} ,

ii. 1 is the multiplicative identity,iii. i2 = j2 = k2 = ijk = −1,iv. ij = −ji = k, jk = −kj = i, ki = −ik = j.

The definition is natural, in the sense that any unital ring homomorphismR1 → R2 extends to a unital ring homomorphism H(R1) → H(R2) bymapping 1 to 1, i to i, j to j and k to k [23, 2.5.1 Definition]. Let α =α1+α2i+α3j+α4k be a quaternion. Here α0 is a real part ,and α2i+α3j+α4k

is an imaginary part. Also, α1, α2, α2, and α2 are components of α quater-nion. Multiplication of quaternions is non-commutative. But, if the imaginaryparts of quaternions are parallel to each other, then multiplication of quater-nions is commutative [14]. Also, multiplication of α and α−1 is commutativesince αα−1 = α−1α = 1.

Definition 2.2. α = α1 + α2i + α3j + α4k is a Hurwitz integer just if eitherα1, α2, α3, α4 ∈ Z or α1, α2, α3, α4 ∈ Z + 1

2 . The set of all Hurwitz integersthat denoted by H (Z) is shown by

H (Z) ={

α1 + α2i + α3j + α4k : α1, α2, α3, α4 ∈ Z or α1, α2, α3, α4 ∈ Z + 12

}

= H(Z)⋃

H(Z + 12 )

(2.1)

The ring of Hurwitz integers H(Z) is forms a subring of the ring of all quater-nions because of closed under multiplication and addition. Let α = α1+α2i+α3j+α4k is a Hurwitz integer. The conjugate of α is α = α1−α2i−α3j−α4k,

the norm of α is N (α) = α · α = α21 + α2

2 + α23 + α2

4, and the inverse of α isα−1 = α

N(α) where its norm is non-zero. The units of H under multiplication


are{

±1,±i,±j,±k, ±1±i±j±k2

}

. So α is a unit of H such that N (α) = 1.

The set has 24 elements.

Definition 2.3. α = α1 + α2i+ α3j + α4k a prime Hurwitz integer just if itsnorm is a rational prime integer.

Example 2.1. α = 2 − 3i + j + 3k and β = 32 + 5

2 i −32j + 7

2k are the

prime Hurwitz integers because of N (α) = 22 + (−3)2 + 12 + 32 = 23 and

N (β) =(

32

)2+(

52

)2+(

− 32

)2+(

72

)2= 92

4 = 23.

Definition 2.4. α = α1 + α2i + α3j + α4k is a primitive Hurwitz integerjust if greater common divisor of its components is equal to 1. That is,gcd (α1, α2, α3, α4) = 1.

Example 2.2. α = 3 + 4i + 2j + k is a primitive Hurwitz integer because ofgcd(3, 4, 2, 1) = 1. But it is not a prime Hurwitz integer because of N(α) =(3)2 + (4)2 + (2)2 + (1)2 = 30.

Definition 2.5. The nearest integer rounding notation denoted by ⌊·⌉ is de-fined as rounding a rational number to the integer closest to its.For quater-nions, each component of a quaternion is separately rounding to the integerclosest to its. So, we obtain Hurwitz integers whose components are in Z froma quaternions. Note that the rounding is done in the direction +∞ in thisstudy.

Example 2.3. Let α = 54 + 1

2 i −12j −

52k ∈ H (R). If we use nearest integer

rounding notation for α, then we obtain a Hurwitz integer whose componentsare in Z. That is,

⌊α⌉ = ⌊ 54 + 1

2 i−12j −

52k⌉

= ⌊ 54⌉+ ⌊ 1

2 i⌉+ ⌊− 12j⌉+ ⌊− 5

2k⌉= 1 + 1 · i+ 0 · j + (−2) · k= 1 + i− 2k.

(2.2)

The residue class set for codes over Gaussian integers that are two-dimensional signal space are constructed by the modulo function technique.Similarity Lipschitz constellation for codes over Lipschitz integers[25], we usethis technique to construct Hurwitz constellation that lies on Hurwitz inte-gers. This technique, known as the modulo function, is given by the followingdefinition. Note that we consider the primitive Hurwitz integers whose normis both a prime integer and not a prime integer, and left residue class set ofprimitive Hurwitz integer in this study.

Definition 2.6. The modulo function µ : ZN(π) → Hπ is defined by

µπ (z) = zmodπ = z − π · ⌊α−1z⌉ = z − π⌊πz

N(π)⌉ (2.3)

where π is a primitive Hurwitz integer and z ∈ ZN(π). Here ZN(π) is thewell-known residual class ring of ordinary integers with N(π) elements, Hπ

is the left residual class set of z modulo π, and µπ (z) is given remainder


of z with respect to modulo π. We can also consider z as a Hurwitz integersuch that its imaginary part is zero where z ∈ Z. The quotient ring of theHurwitz integers modulo this equivalence relation, which we denote as Hπ ={

zmodπ : z ∈ ZN(π)

}

. The Hπ set contains N(π) elements. If π is a primeHurwitz integer, then the modulo function µ defines a bijective mapping fromZN(π) intoHπ which is a four-dimensional signal space. Therefore, the modulofunction µ is a ring isomorphism between ZN(π) and Hπ. Because there existsa inverse map [8] and we have µ(z1 + z2) = µ(z1) + µ(z2) and µ(z1z2) =µ(z1)µ(z2) for any z1, z2 ∈ ZN(π). If π is a primitive Hurwitz integer, themodulo function µ is a group isomorphism with respect to addition betweenZN(π) andHπ. Because there exists a inverse map [8] and we have µ(z1+z2) =µ(z1)+µ(z2) for any z1, z2 ∈ ZN(π). After we define encoder Hurwitz integersset in the following section, we can define a ring isomorphism between ZN(π)

and Hπ where π is an encoder Hurwitz integer.

In engineering, the ”signal constellation” has been used as a communica-tion term. In mathematics, the ”signal constellation” means for residue classset. In the rest of this study, we use the” signal constellation” term insteadof the” left residue class set” term. You can find more details which relatedto the arithmetics properties about arithmetic properties of quaternions andHurwitz integers in [23-24].

3. 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 symbolfor absolute value. If we generalize the Euclid division algorithm for Hur-witz integers whose components are in Z, then there exists unique Hurwitzintegers β and γ such that α = πβ + γ, 0 ≤ N(γ) ≤ N(π) where α, π ∈ Hsuch that their components are in Z. Therefore, the Euclid division algorithmdoes not work for Hurwitz integers whose components are in Z because of0 ≤ N(γ) ≤ N(π). So, the primitive Hurwitz integers whose components arein Z do not have the ”division with small remainder” property. If we gener-alize the Euclid division algorithm for Hurwitz integers whose componentsare in Z + 1

2 , then there exists unique Hurwitz integers β and γ such thatα = πβ+γ, 0 ≤ N(γ) < N(π) where α, π ∈ H such that their components arein Z+ 1

2 . Therefore, the Euclid division algorithm works for Hurwitz integers

whose components are in Z+ 12 because of 0 ≤ N(γ) < N(π). So, the primi-

tive Hurwitz integers whose components are in Z+ 12 have the ”division with

small remainder” property[24]. Also, the Euclid division algorithm generallyworks for prime Hurwitz integers. Because each element in the Hπ has theminimal norm. So, the prime Hurwitz integers have the ”division with smallremainder” property. The following proposition and lemma imply that prim-itive Hurwitz integers whose components are in Z do not have the ”division


with small remainder” property since not working Euclid division algorithmfor primitive Hurwitz integers whose each component is an odd integer.

Proposition 3.1. Let π is a primitive Hurwitz integer whose each componentis an odd integer. Then,

N(µπ(N(π)

2)) = N(π) (3.1)

with respect to equation (2.3).

Proof. Let π = π1+π2i+π3j+π4k is a Hurwitz integer whose each componentare an odd integer such that gcd(π1, π2, π3, π4) = 1. The conjugate of π is

π = π1−π2i−π3j−π4k. We show that is N(µπ(N(π)

2 )) = N(π) with respectto equation (2.3).

µπ(N(π)

2 ) = N(π)2 − π⌊

πN(π)2

N(π) ⌉

= N(π)2 − π⌊π

2 ⌉

= N(π)2 − (π1 + π2i+ π3j + π4k)⌊

π1−π2i−π3j−π4k2 ⌉.

(3.2)

⌊π1

2 ⌉ = ((π1

2 ) + 12 ), ⌊

−π2

2 ⌉ = ((−π2

2 ) + 12 ), ⌊

−π3

2 ⌉ = ((−π3

2 ) + 12 ) and

⌊−π4

2 ⌉ = ((−π4

2 ) + 12 ) since π1, π2, π3 and π4 are odd integers. Then,

µπ(N(π)

2 ) = N(π)2 − (π1 + π2i+ π3j + π4k)[(

π1

2 + 12 ) + (−π2

2 + 12 )i

+(−π3

2 + 12 )j + (−π4

2 + 12 )k]

= N(π)2 − (

π21+π1

2 + (−π1π2+π1

2 )i+ (−π1π3+π1

2 )j

+(−π1π4+π1

2 )k + (π2π1+π2

2 )i+π22−π2

2 + (−π2π3+π2

2 )k

+(π2π4−π2

2 )j + (π3π1+π3

2 )j + (π3π2−π3

2 )k + (π23−π32 )

+(−π3π4+π3

2 )i+ (π4π1+π4

2 )k + (−π4π2+π4

2 )j

+(π4π3−π4

2 )i+π24−π4

2

= N(π)2 − [

π21+π2

2+π23+π2

4+π1−π2−π3−π4

2+(−π1π2+π1+π2π1+π2−π3π4+π3+π4π3−π4

2 )i+(−π1π3+π1+π2π4−π2+π3π1+π3−π4π2+π4

2 )j+(−π1π4+π1−π2π3+π2+π3π2−π3+π4π1+π4

2 )k]

µπ(N(π)

2 ) = N(π)2 − N(π)

2 + π1−π2−π3−π4

2 + (π1+π2+π3−π4

2 )i+(π1−π2+π3+π4

2 )j + (π1+π2−π3+π4

2 )k= π1−π2−π3−π4

2 + (π1+π2+π3−π4

2 )i+ (π1−π2+π3+π4

2 )j+(π1+π2−π3+π4

2 )k.

(3.3)

Therefore, we have

N(µπ(N(π)

2 )) = N(π1−π2−π3−π4

2 + (π1+π2+π3−π4

2 )i+(π1−π2+π3+π4

2 )j + (π1+π2−π3+π4

2 )k)

=4(π2

1+π22+π2

3+π24)

4= π2

1 + π22 + π2

3 + π24

(3.4)


Since N(π) = π21 + π2

2 + π23 + π2

4 , we have

N(µπ(N(π)

2)) = N(π). (3.5)

This completes the proof. �

Let γ = µπ(N(π)

2 ). By proposition 3.1, we have N(µπ(N(π)

2 )) = N(γ) =N(π). In other words, the norm of the remainder is equal to the norm of thedivisor. We generalize proposition 3.1 for primitive Hurwitz integers whosenorm is not a prime integer where its each component is in Z with the fol-lowing 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

N(µπ(α)) ≤ N(π) (3.6)

with respect to equation (2.3).

Proof. Let π be a primitive Hurwitz integer, and α = α1 + α2i+ α3j + α4k

be a Hurwitz integers which its norm is non-zero. If π is a primitive Hurwitzinteger whose all component are in Z, by equation (2.3)

µπ(α) = α− π⌊ παN(π)⌉

π−1µπ(α) = π−1α− π−1π⌊παππ

⌉= π−1α− ⌊α

π⌉

= π−1(α1 + α2i+ α3j + α4k)− ⌊α1+α2i+α3j+α4kπ

⌉= π−1α1 + π−1α2i+ π−1α3j + π−1α4k

−⌊π−1α1 + π−1α2i+ π−1α3j + π−1α4k⌉= π−1α1 + π−1α2i+ π−1α3j + π−1α4k

−⌊π−1α1 + π−1α2i+ π−1α3j + π−1α4k⌉.= π−1α1 − ⌊π−1α1⌉+ (π−1α2 − ⌊π−1α2⌉)i

+(π−1α3 − ⌊π−1α3⌉)j + (π−1α4 − ⌊π−1α4⌉)k.

(3.7)

Therefore, we have

| π−1α1 − ⌊π−1α1⌉ | ≤ 12

| π−1α2 − ⌊π−1α2⌉ | ≤ 12

| π−1α3 − ⌊π−1α3⌉ | ≤ 12

| π−1α4 − ⌊π−1α4⌉ | ≤ 12 .

(3.8)

Hereby,

(π−1α1 − ⌊π−1α1⌉)2 ≤ ( 12 )

2

(π−1α2 − ⌊π−1α2⌉)2 ≤ ( 12 )

2

(π−1α3 − ⌊π−1α3⌉)2 ≤ ( 12 )

2

(π−1α4 − ⌊π−1α4⌉)2 ≤ ( 12 )

2

(3.9)


So, we have

N(π−1µπ(α)) = (π−1α1 − ⌊π−1α1⌉)2 + (π−1α2 − ⌊π−1α2⌉)

2

+(π−1α3 − ⌊π−1α3⌉)2 + (π−1α4 − ⌊π−1α4⌉)

2

≤ ( 12 )2 + ( 12 )

2 + ( 12 )2 + ( 12 )

2

N(π−1)N(µπ(α)) ≤ 14 + 1

4 + 14 + 1

41

N(π)N(µπ(α)) ≤ 1

N(µπ(α)) ≤ N(π).(3.10)


Consequently, primitive Hurwitz integers, whose norm is not a primeinteger, such that each component is in Z, do not have the ”division withsmall remainder” property. So, the Euclid division algorithm does not workfor primitive Hurwitz integers, whose norm is not a prime integer, such thateach component is in Z. With the following example, given practices forproposition 3.1.

Example 3.1. Let π = 3+i+j+3k be a Hurwitz integer. Also, π is a primitiveHurwitz integer because of gcd(3, 1, 1, 3) = 1. By equation (2.3), π Hurwitzconstellation is

Hπ =

µπ (0) = 0, µπ (1) = 1, µπ (2) = 2, µπ (3) = 3,

µπ (4) = −2− 2j, µπ (5) = −1− 2j, µπ (6) = −2j,

µπ (7) = 1− 2j, µπ (8) = 2− 2j, µπ (9) = 3− 2j,

µπ (10) = 1− i− 3j − 3k, µπ (11) = −3 + 2j,

µπ (12) = −2 + 2j, µπ (13) = −1 + 2j, µπ (14) = 2j,

µπ (15) = 1 + 2j, µπ (16) = 2 + 2j, µπ (17) = −3,

µπ (18) = −2, µπ (19) = −1

. (3.11)

This set contains 20 elements because of N(π) = 32+12+12+32 = 20. Also,N(µπ(10)) = N(π) because of N(µπ(10)) = 32 + (1)2 + (1)2 + (3)2 = 20.The norm of other elements in the set (3.11) is less than the norm of π.

Consequently, π = 3 + i + j + 3k Hurwitz integer does not have the ”di-vision with small remainder” property. In addition, to be a Euclidean met-ric, the inequality d(x

′

, y′

) + d(y′

, z′

) ≥ d(x′

, z′

) should be verified. Because

the inequalities i) d(x′

, y′

) = 0 if and only if x′

= y′

where x′

, y′

∈ Hπ,

and ii) d(x′

, y′

) = d(y′

, x′

) where x′

, y′

∈ Hπ, are supplied. We consider

x′

= 1 − i − 3j − 3k, y′

= −1 and z′

= −2 in (3.11). d(x′

, y′

) = 23since N(y − x) = N(−1 − 1 + i + 3j + 3k) = N(−2 + i + 3j + 3k) = 23,

d(y′

, z′

) = 1 since N(z − y) = N(−2 + 1) = N(−1) = 1, and d(x′

, z′

) = 28since N(z−y) = N(−2−1+ i+3j+3k) = N(−3+ i+3j+3k) = 28. Hereby,we have

d(x′

, y′

) + d(y′

, z′

) ≥ d(x′

, z′

)23 + 1 ≥ 2824 ≥ 28.

(3.12)


But this is not true. Consequently, the Euclidean metric is not provide forthe Hπ constellation constructed by π = 3+ i+ j + 3k Hurwitz integer, andthe Euclidean division algorithm is not work for π = 3 + i+ j + 3k Hurwitzinteger.

We define a set that consists of the primitive Hurwitz integers that havethe ”division with small remainder” property with the following definition.This set is a subset of the primitive Hurwitz integers, and Hurwitz integers.

Definition 3.1. Let α = α1+α2i+α3j+α4k be a Hurwitz integer. α Hurwitzinteger is called an encoder Hurwitz integer if it is satisfying the followingconditions.If α1, α2, α3, α4 ∈ Z,

• α1, α2, α3, α4 ∈ Z+,

• α1, α2, α3, α4 are not same parity, i.e. α1, α2, α3 and α4 all togetherneither even nor odd integers,

• gcd (α1, α2, α3, α4) = 1.

Or,

• α1, α2, α3, α4 ∈ (Z ∪ {0}) + 12 ,

In other words, α = α1+α2i+α3j+α4k primitive Hurwitz integer such thatits all components are not odd integers, or α1, α2, α3, α4 ∈ (Z ∪ {0}) + 1

2 , iscalled an encoder Hurwitz integer.

The definition 3.1 is a flexible definition. Namely, the elements of theencoder Hurwitz integers set are expandable or collapsible depending on theused modulo technique. In this study, we defined the above definition withrespect to the modulo function defined in definition 2.6. Let now us showthat the modulo function µ defined between ZN(π) and Hπ by equation (2.3)is a ring isomorphism with the following theorems.

Theorem 3.1. Let π be an encoder Hurwitz integer, and z1, z2 ∈ ZN(π). µ :ZN(π) → Hπ modulo function is a ring homomorphism with respect to µπ(z1+z2) = (µπ(z1) + µπ(z2))modπ and µπ(z1z2) = (µπ(z1)µπ(z2))modπ.

Proof. Let π be an encoder Hurwitz integer and, z1, z2 ∈ ZN(π). Define µ :

ZN(π) → Hπ modulo function by µπ(z) = zmodπ = z − π⌊ πzN(π)⌉. Note

that each component of λ1 and λ2 are in Z with respect to round notation.Therefore,

µπ(z1 + z2) = (z1 + z2)modπ.

= z1 + z2 − π⌊π(z1+z2)N(π) ⌉.

(3.13)

If µπ(z1) = z1modπ and µπ(z2) = z2modπ, then, there exists λ1 and λ2 Hur-witz integers such that z1 = πλ1+µπ(z1) and z2 = πλ2+µπ(z2), respectively.


Then, we have

µπ(z1 + z2) = πλ1 + µπ(z1) + πλ2 + µπ(z2)

−π⌊π(πλ1+µπ(z1)+πλ2+µπ(z2))N(π) ⌉

= πλ1 + µπ(z1) + πλ2 + µπ(z2)

−π⌊ππλ1+πµπ(z1)+ππλ2+πµπ(z2)N(π) ⌉

= πλ1 + µπ(z1) + πλ2 + µπ(z2)

−π⌊λ1 + λ2 +πµπ(z1)+πµπ(z2)

N(π) ⌉.

(3.14)

⌊λ1 + λ2⌉ = λ1 + λ2 since λ1 and λ2 are the Hurwitz integers whose eachcomponent is in Z. Then, we have

µπ(z1 + z2) = πλ1 + µπ(z1) + πλ2 + µπ(z2)− πλ1 − πλ2

−π⌊πµπ(z1)+πµπ(z2)N(π) ⌉

= µπ(z1) + µπ(z2)− π⌊π(µπ(z1)+µπ(z2))N(π) ⌉.

(3.15)

So, we have

µπ(z1 + z2) = (µπ(z1) + µπ(z2))modπ. (3.16)

On the other hand,

µπ(z1z2) = z1z2modπ

= z1z2 − π⌊π(z1z2)N(π) ⌉.

(3.17)

If µπ(z1) = z1modπ and µπ(z2) = z2modπ, then, there exists λ1 and λ2 Hur-witz integers such that z1 = πλ1+µπ(z1) and z2 = πλ2+µπ(z2), respectively.Then, we have

µπ(z1z2) = (πλ1 + µπ(z1))(πλ2 + µπ(z2))

−π⌊π(πλ1+µπ(z1))(πλ2+µπ(z2))N(π) ⌉

= πλ1πλ2 + πλ1µπ(z2) + µπ(z1)πλ2 + µπ(z1)µπ(z2)

−π⌊ππλ1πλ2+ππλ1µπ(z2)+πµπ(z1)πλ2+πµπ(z1)µπ(z2)N(π) ⌉.

Since N(π) = ππ, we have

µπ(z1z2) = πλ1πλ2 + πλ1µπ(z2) + µπ(z1)πλ2 + µπ(z1)µπ(z2)

−π⌊λ1πλ2 + λ1µπ(z2) +πµπ(z1)πλ2

N(π) + πµπ(z1)µπ(z2)N(π) ⌉.

Since λ1πλ2, λ1µπ(z2) andπµπ(z1)πλ2

N(π) are Hurwitz integers whose each com-

ponent is in Z, we have

µπ(z1z2) = πλ1πλ2 + πλ1µπ(z2) + µπ(z1)πλ2 + µπ(z1)µπ(z2)− πλ1πλ2

−πλ1µπ(z2)− ππµπ(z1)πλ2

N(π) − π⌊πµπ(z1)µπ(z2)N(π) ⌉

= µπ(z1)µπ(z2)− π⌊πµπ(z1)µπ(z2)N(π) ⌉.

So, µπ(z1z2) = µπ(z1)µπ(z2) modπ. Consequently, µ function is a ring homo-morphism. This completes this proof. �

Theorem 3.2. Let π be an encoder Hurwitz integer. Then,

ZN(π)∼= Hπ


with respect to equality (2.3).

Proof. Let π be an encoder Hurwitz integer and, z1, z2 ∈ ZN(π). The modulofunction defines a mapping from ZN(π) to Hπ. This mapping is µ : ZN(π) →Hπ :

µπ(z) = zmodπ = γ = z − π⌊πz

N(π)⌉. (3.18)

According to theorem 3.1, µ function is a ring homomorphism. This mappingis a surjective ring homomorphism because of Imµ = {µπ(z) : z ∈ ZN(π)} =Hπ. If z = 0 where z ∈ ZN(π), then we have

µπ(0) = 0− π⌊ π0N(π)⌉

= 0− π⌊0⌉= 0− π0= 0.

(3.19)

If z 6= 0 where z ∈ ZN(π), then µπ(z) is to greater or equal than 1. Hereby,this mapping is a bijective ring homomorphism because of Kerµ = {z ∈ZN(π) : µπ(z) = 0} = {z ∈ ZN(π) : z = 0} = {0}. µ function is a ringisomorphism since it is both a surjective ring homomorphism and a bijectivering homomorphism, i.e. ZN(π)

∼= Hπ. �

The following proposition demonstrates that the encoder Hurwitz inte-gers have the ”division small remainder” property.

Proposition 3.2. Let π is an encoder Hurwitz integer whose each componentis not an odd integer. Then,

N(µπ(z)) < N(π). (3.20)

Proof. We shall analyze and prove this theorem case by case. Let π = π1 +π2i+ π3j + π4k is an encoder Hurwitz integer.Case : 1 Let π is an encoder Hurwitz integer such that π1 is an even integer,and π2, π3 and π4 are odd integers. So, N(π) is a odd integer. By equation(2.3),

µπ(z) = z − π⌊π−1z⌉ (3.21)

Hereby,

µπ(z)π−1 = π−1z − π−1π⌊π−1z⌉

= π−1z − ⌊π−1z⌉= πz

N(π) − ⌊ πzN(π)⌉.

(3.22)

Since π = π1 − π2i− π3j − π4k, we have

µπ(z)π−1 = π1z

N(π) − ( π2zN(π) )i− ( π3z

N(π) )j − ( π4zN(π) )k

−⌊ π1zN(π)⌉ − ⌊ π2z

N(π)⌉i− ⌊ π3zN(π)⌉j − ⌊ π4z

N(π)⌉k

= π1zN(π) − ⌊ π1z

N(π)⌉ − ( π2zN(π) − ⌊ π2z

N(π)⌉)i

−( π3zN(π) − ⌊ π3z

N(π)⌉)j − ( π4zN(π) − ⌊ π4z

N(π)⌉)k

(3.23)


Since π1 is an even integer, π2, π3 and π2 are odd integers, N(π) is an odd

integer, and N(π)2 is not a integer,

0 ≤ | π1zN(π) − ⌊ π1z

N(π)⌉ | < 12

0 ≤ | π2zN(π) − ⌊ π2z

N(π)⌉ | < 12

0 ≤ | π3zN(π) − ⌊ π3z

N(π)⌉ | < 12

0 ≤ | π4zN(π) − ⌊ π4z

N(π)⌉ | < 12 .

(3.24)

Therefore, we have

N( π1zN(π) − ⌊ π1z

N(π)⌉) +N( π2zN(π) − ⌊ π2z

N(π)⌉) +N( π3zN(π) − ⌊ π3z

N(π)⌉)

+N( π4zN(π) − ⌊ π4z

N(π)⌉) < ( 12 )2 + ( 12 )

2 + ( 12 )2 + ( 12 )

2 = 1.(3.25)

Hereby, we have

N(µπ(z)π−1) = N(µπ(z))N(π−1) = N(µπ(z))

1

N(π)< 1. (3.26)

Consequently,

N(µπ(z)) < N(π). (3.27)

Case : 2 Let π is an encoder Hurwitz integer such that π1 and π2 are evenintegers, and π3 and π4 are odd integers. Firstly, we should check whetherto verify or not equation 3.1 in preposition 3.1 in case of N(π) is an even

integer. Let z = N(π)2 . By equation (2.3), we have

µπ(N(π)

2 ) = N(π)2 − π⌊π−1N(π)

2 ⌉. (3.28)

Hereby,

µπ(N(π)

2 )π−1 = π−1N(π)2 − π−1π⌊π−1N(π)

2 ⌉= π

2 − ⌊π2 ⌉

(3.29)

Since π = π1 − π2i− π3j − π4k,

µπ(N(π)

2 )π−1 = π1

2 − (π2

2 )i− (π3

2 )j − (π4

2 )k−⌊π1

2 ⌉+ ⌊π2

2 ⌉i+ ⌊π3

2 ⌉j + ⌊π4

2 ⌉k= π1

2 − ⌊π1

2 ⌉ − (π2

2 − ⌊π2

2 ⌉)i−(π3

2 − ⌊π3

2 ⌉)j − (π4

2 − ⌊π4

2 ⌉)k

(3.30)

⌊π1

2 ⌉ = π1

2 and ⌊π2

2 ⌉ = π2

2 since π2 and π3 are even integers. Also, ⌊π3

2 ⌉ = π3+12

and ⌊π4

2 ⌉ = π4+12 since π3 and π4 are odd integers. So,

µπ(N(π)

2 )π−1 = π1

2 − π1

2 − (π2

2 − π2

2 )i−(π3

2 − π3+12 )j − (π4

2 − π4+12 )k

= ( 12 )j + ( 12 )k.

(3.31)

Hereby, we have

N(µπ((N(π)

2 )π−1) = N(( 12 )j + ( 12 )k)

N(µπ(N(π)

2 ))N(π−1) = ( 12 )2 + ( 12 )

2

N(µπ(N(π)

2 ))( 1N(π) ) = 1

4 + 14

N(µπ(N(π)

2 )) = N(π)2 .

(3.32)


Therefore,

N(µπ(N(π)

2)) < N(π). (3.33)

Let z 6= N(π)2 . By equation (2.3), we have

µπ(z) = z − π⌊π−1z⌉. (3.34)

Hereby,

µπ(z)π−1 = π−1z − π−1π⌊π−1z⌉

= π−1z − ⌊π−1z⌉= πz

N(π) − ⌊ πzN(π)⌉.

(3.35)



N(π) − ( π2zN(π) )i− ( π3z

N(π) )j − ( π4zN(π) )k

−⌊ π1zN(π)⌉ − ⌊ π2z

N(π)⌉i− ⌊ π3zN(π)⌉j − ⌊ π4z

N(π)⌉k

= π1zN(π) − ⌊ π1z

N(π)⌉ − ( π2zN(π) − ⌊ π2z

N(π)⌉)i

−( π3zN(π) − ⌊ π3z

N(π)⌉)j − ( π4zN(π) − ⌊ π4z

N(π)⌉)k.

(3.36)

Since z 6= N(π)2 ,

0 ≤ | π1zN(π) − ⌊ π1z

N(π)⌉ | < 12

0 ≤ | π1zN(π) − ⌊ π1z

N(π)⌉ | < 12

0 ≤ | π3zN(π) − ⌊ π3z

N(π)⌉ | < 12

0 ≤ | π4zN(π) − ⌊ π4z

N(π)⌉ | < 12 .

(3.37)

Hereby, we have

N( π1zN(π) − ⌊ π1z

N(π)⌉) +N( π2zN(π) − ⌊ π2z

N(π)⌉) +N( π3zN(π) − ⌊ π3z

N(π)⌉)

+N( π4zN(π) − ⌊ π4z

N(π)⌉) < ( 12 )2 + ( 12 )

2 + ( 12 )2 + ( 12 )

2 = 1(3.38)

Therefore,


1

N(π)< 1 (3.39)

Consequently,

N(µπ(z)) < N(π). (3.40)

Case : 3 Let π is an encoder Hurwitz integer such that π1, π2 and π3 are evenintegers, and π4 is an odd integer. So, N(π) is an odd integer. By equation(2.3), we have

µπ(z) = z − π⌊π−1z⌉ (3.41)

Hereby,

µπ(z)π−1 = π−1z − π−1π⌊π−1z⌉

= π−1z − ⌊π−1z⌉= πz

N(π) − ⌊ πzN(π)⌉.

(3.42)




N(π) − ( π2zN(π) )i− ( π3z

N(π) )j − ( π4zN(π) )k

−⌊ π1zN(π)⌉ − ⌊ π2z

N(π)⌉i− ⌊ π3zN(π)⌉j − ⌊ π4z

N(π)⌉k

= π1zN(π) − ⌊ π1z

N(π)⌉ − ( π2zN(π) − ⌊ π2z

N(π)⌉)i

−( π3zN(π) − ⌊ π3z

N(π)⌉)j − ( π4zN(π) − ⌊ π4z

N(π)⌉)k

(3.43)

Since π4 is an odd integer, π1, π2 and π3 are even integers, and N(π) is anodd integer,

0 ≤ | π1zN(π) − ⌊ π1z

N(π)⌉ | < 12

0 ≤ | π2zN(π) − ⌊ π2z

N(π)⌉ | < 12

0 ≤ | π3zN(π) − ⌊ π3z

N(π)⌉ | < 12

0 ≤ | π4zN(π) − ⌊ π4z

N(π)⌉ | < 12 .

(3.44)

Hereby, we have

N( π1zN(π) − ⌊ π1z

N(π)⌉) +N( π2zN(π) − ⌊ π2z

N(π)⌉) +N( π3zN(π) − ⌊ π3z

N(π)⌉)

+N( π4zN(π) − ⌊ π4z

N(π)⌉) < ( 12 )2 + ( 12 )

2 + ( 12 )2 + ( 12 )

2 = 1(3.45)

Therefore,


1

N(π)< 1 (3.46)

Consequently,

N(µπ(z)) < N(π). (3.47)


With the following examples, giving an example for each case in theproposition 3.2.

Example 3.2. Case 1 π = 1 + 3i+ 2j + k is an encoder Hurwitz integer. Byequation (2.3), the π Hurwitz constellation is

Lπ =

µπ (0) = 0, µπ (1) = 1, µπ (2) = 2, µπ (3) = i + j − 2k,

µπ (4) = −1 + 2j + k, µπ (5) = 2j + k, µπ (6) = 1 + 2j + k,

µπ (7) = 2 + 2j + k, µπ (8) = −2 − 2j − k, µπ (9) = −1 − 2j − k,

µπ (10) = −2j − k, µπ (11) = 1 − 2j − k, µπ (12) = −i − j + 2k,

µπ (13) = −2, µπ (14) = −1,

. (3.48)

The set contains 15 elements because of N(π) = 12 + 32 + 22 + 12 = 15. Thenorm 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 Hurwitz integer. Byequation (2.3), the π Hurwitz constellation is

Lπ =

µπ (0) = 0, µπ (1) = 1, µπ (2) = 2, µπ (3) = 3,

µπ (4) = 1 + 2i + 2j − k, µπ (5) = −2 − 2j − k, µπ (6) = −1 − 2j − k,

µπ (7) = −2j − k, µπ (8) = 1 − 2j − k, µπ (9) = 2 − 2j − k,

µπ (10) = −1 + 2j + k, µπ (11) = 2j + k, µπ (12) = 1 + 2j + k,

µπ (13) = 2 + 2j + k, µπ (14) = −1 − 2i − 2j + k,

µπ (15) = −2i − 2j + k, µπ (16) = −2, µπ (17) = −1

. (3.49)



Example 3.4. Case 3 π = 2+ 3i+ 2j + 2k is an encoder Hurwitz integer. Byequation (2.3), the π Hurwitz constellation is

Lπ =

µπ (0) = 0, µπ (1) = 1, µπ (2) = 2, µπ (3) = 3,

µπ (4) = 1 + 2i + 2j − 2k, µπ (5) = 2 + 2i + 2j − 2k,

µπ (6) = 3 − i − j + k, µπ (7) = −2 − i − j + k,

µπ (8) = −1 − i − j + k, µπ (9) = −i − j + k,

µπ (10) = 1 − i − j + k, µπ (11) = −1 + i + j − k,

µπ (12) = i + j − k, µπ (13) = 1 + i + j − k,

µπ (14) = 2 + i + j − k, µπ (15) = 3 + i + j − k,

µπ (16) = −2 − 2i − 2j + 2k, µπ (17) = −1 − 2i − 2j + 2k,

µπ (18) = −3, µπ (19) = −2, µπ (17) = −1

. (3.50)


Example 3.2, example 3.3, and example 3.4 are verified all the conditionsfor it to be an Euclidean metric. Also the Euclidean division algorithm worksfor the Hurwitz integers in example 3.2, example 3.3, and example 3.4. As aresult of these examples, we represent the following proposition.

In the following example, we show that it does not have the ”divisionwith small remainder” property of a Hurwitz integer used to obtain the Hur-witz constellation constructed with a different technique by Rohweder et al..

Example 3.5. In [19], Rohweder et al. presented the new construction methodfor Hurwitz integers by

µπ(z) = z − ⌊zπ

N(π)⌉π (3.51)

where π is a primitive Hurwitz integer and z ∈ ZN(π). They proposed four-dimensional Hurwitz integer signal constellations are obtained from the fol-lowing mapping

Hπ = Lπ

⋃

Oπ, (3.52)

where Lπ is the subset of Lipschitz integers, which are quaternions whose allcomponents are in Z, which can be evaluated by

Lπ = {µπ(a+ bj) : a, b ∈ ZN(π)} (3.53)

where ZN(π) denotes the ring of integers modulo N(π). Also, Oπ in equation(3.52) is the corresponding coset of half-integers, which can be calculated by

Oπ = {µπ(h+ w) : h ∈ Lπ}, (3.54)

where w = 12 +

12 i+

12j+

12k. This set must be 100 elements. The proposition

3.1 says that if π is a Hurwitz integer whose each component is an odd integer

and, N(π) is an even integer, then we should check the N(π)2 integer at the

construction method used for Hurwitz integers. We consider 3 + i in [19,


Table I]. Hereby, N(π) = 32 + 12 = 10. Let be a = N(π)2 = 10

2 = 5 and

b = N(π)2 = 10

2 = 5. So, we consider µπ(5 + 5j) in Lπ. By equation (3.51),

µπ(5 + 5j) = 5 + 5j − ⌊ (5+5j)(3−i)10 ⌉(3 + i)

= 5 + 5j − ⌊ 15−5i+15j+5k10 ⌉(3 + i)

= 5 + 5j − (2 + 2j + k)(3 + i)= 5 + 5j − 6− 2i− 7j − k

= −1− 2i− 2j − k.

(3.55)

Hereby, N(µπ(5 + 5j)) = N(π) because of N(µπ(5 + 5j)) = (−1)2 + (−2)2 +(−2)2+(−1)2 = 10. So, 5+5j ≡ 0mod(3+i). If a = 0 and b = 0, then µπ(0) =0. Thereby, the Lπ has elements less than 100 elements since µπ(0) = µπ(5+5j) ≡ 0mod(3+ i). This contradicts to the size of the Oπ constellation, whichis with 100 elements. Similarly, we can not construct Oπ constellation with100 elements by 2+2i+ j+ k primitive Lipschitz integer, too. Consequently,we can say that 3+ i and 2+ 2i+ j + k do not have the ”division with smallremainder ” property with respect to the method in [19]. The method in [19]inappropriate to construct the Hurwitz constellation with 100 elements.

The technique in [19] is more appropriate for the Lipschitz integerswhose norm is an odd number but inappropriate for the Lipschitz integerswhose norm is an even number. Note that the definition 3.1 is a flexible def-inition. According to the modulo technique in [19], we can define the set ofthe encoder Hurwitz integers with ”The Hurwitz integer whose the greatestcommon divisor of its components is one and its norm is an odd number iscalled an encoder Hurwitz integer.”. The set of encoder Hurwitz integers isthe set of the Hurwitz integers remaining with taking out the Hurwitz integersproviding proposition 3.1 from the Hurwitz integers set, in general. We referto proposition 3.1 and example 3.5 for this general definition. Note that weconsider the definition 3.1 in this study. Note that we consider definition 3.1in this study. We can come to a conclusion that the Euclid division algorithmworks for the elements of the encoder Hurwitz integers set. So, we can con-struct well-defined Hurwitz constellations in terms of algebraic constructionsfor codes over Hurwitz integers. Consequently, we should use proposition 3.1to check whether the Hurwitz integers used to construct Hurwitz constella-tions have the ”division with small remainder” property or not. Consequently,we should use proposition 3.1 to check whether the Hurwitz integers used toconstruct Hurwitz constellations have the ”division with small remainder”property or not.

4. 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 lieson encoder Hurwitz integers for transmission over AWGN channel by agency


of average energy, CFM, and SNR gains. Note that we investigate the perfor-mances of Hurwitz constellations constructed with Hurwitz integers whosecomponents are in Z+ 1

2 by using the technique used for Gaussian constella-tions in [9] and Lipschitz integers in [25] in this study. Because the Hurwitzconstellations constructed of primitive Hurwitz integers which their compo-nents are in Z show the same performances with Lipschitz constellationsin [25]. Therefore, we give set partitioning property on larger Hurwitz inte-gers namely, proposed Hurwitz integers, since the Hurwitz constellations thathave the same size with Gaussian constellations are almost shown the sameperformances for transmission over the AWGN channel. We follow the pro-cedures in [25] for some distance, performance measures and set partitioningproperty. The average energy of a constellation denoted by Eπ is computedby

Eπ =1

N (π)

N(π)−1∑

z=0

N (µπ (z)). (4.1)

The squared Euclidean distance of two Hurwitz integers is defined as

dE (α, β) = N (β − α) (4.2)

and the minimum squared Euclidean distance of the constellation is

δ2λ = minα 6=β

dE (α, β) . (4.3)

where α, β ∈ Hπ. In [26], Forney and Wei proposed the constellation figureof merit (CFM) to compare signal constellations of different dimensions. TheCFM is the ratio of the minimum squared Euclidean distance and the averageenergy per two-dimensions. So, the CFM of a M−dimensional constellationis computed by

CFM =Mδ2π2Eπ

. (4.4)

A higher CFM leads to a better performance for transmission over an AWGNchannel [24]. Asymptotic coding gain means for higher signal to noise ratio(SNR) [9]. The SNR of M−dimensional constellation is computed by

SNR = −10 · log10(CFM of signal constellation). (4.5)

The SNR gains of a Hurwitz constellation over the AWGN channel is

SNR = −10 · log10(CFM of Hurwitz signal constellation

CFM of Gaussian noise constellation). (4.6)

Note that the number of elements of the Hurwitz constellation and Gaussianconstellation should be the same to compare performances over the AWGNchannel. A residue class ring of Hurwitz integers Hπ arises from the residueclass ring of integers ZN(π) = {0, 1, . . . , N(π)− 1} for an integer N(π). IfN(π) is not a prime integer, then we can partition the set ZN(π) into subsetsof equal size. Let N = c · d where N is the elements number of Hurwitz

constellation. We can partition the setHπ into c subsetsH(0)π , . . . ,H

(c−1)π each


with d elements. The subsets correspond to the integer sets Z(0)π , . . . ,Z

(c−1)π ,

whereZ(0)N(π) = {0, c, 2c, . . . , (d− 1)c} (4.7)

and Z(1)π , . . . ,Z

(c−1)π are the cosets of Z

(0)π , i.e. Z

(l)π =

{

z : z − l ∈ Z(0)π

}

. Note

that the number of elements of the Hurwitz constellation and Gaussian con-stellation should be the equal size to compare performances over the AWGNchannel but proposed Hurwitz constellations should not be. Hence, we canapply set partitioning property on proposed primitive Hurwitz constellations.The SNR gains of a proposed Hurwitz constellation over the AWGN channelis computed by

SNR = −10 · log10(CFM of H(0)

π

CFM of Gaussian constellation )+10 · log10(

CFM of Hurwitz constellationCFM of Gaussian constellation )

= −10(log10(CFM of H(0)π )

− log10(CFM of Hurwitz constellation))+10 · log10(CFM of Gaussian constellation)−10 · log10(CFM of Gaussian constellation)

= −10(log10(CFM of H(0)π )

− log10(CFM of Hurwitz constellation))

= −10 · log10(CFM of H(0)

π

CFM of Hurwitz constellation )

(4.8)

where Hurwitz signal constellation H(0)π and the Hurwitz constellation are

the equal size. Guzeltepe [18], and Rohweder et al. [19] separately presenteddifferent techniques for Hurwitz constellations. Guzeltepe [18] investigatedperformances of the Hurwitz constellations with N(π)2 elements where π isa primitive Hurwitz integer, over the AWGN channel by using isomorphismbetween Hπ2 and Zπ2 . You can see example 3.5, or [19] for the technique ofRohweder et al.. Note that we use isomorphism between Hπ and Zπ in thisstudy.

Example 4.1. In Table I, we present the performance of the Hurwitz constel-lation constructed by encoder Hurwitz integers whose each component is inZ+ 1

2 over the AWGN channel by means of average energy, CFM, and SNRcoding gains. In Table I, the Hurwitz constellations obtained from the modulofunction technique in this study have almost similar properties as Lipschitzconstellations in the paper of Freudenberger et al. in [25]. The performance ofHurwitz constellations over the AWGN channel in Table I is not so good butbetter than nothing with respect to the performance of the Hurwitz constel-lations 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 constellationswhose components is in Z and the performances of Lipschitz constellationsare the same. Similarity, the performances of proposed Hurwitz constellationswhose components is in Z are the same with the performances of proposedLipschitz constellations in [25, Table I].


Table 1. Table of CFM, Energy and SNR of Hurwitz constellationsconstructed by Hurwitz integers whose coefficients are in Z+ 1

2( d:The

number of elements (norm) of Hurwitz signal constellation )

Constellations

CFM Energy

Primitive Hurwitz Gauss Hurwitz Gauss Hurwitz SNR

d Integer gain

5 3

2+ 3

2i+ 1

2j + 1

2k 1.2500 1.6667 0.8000 1.200 1.25

13 5

2+ 3

2i+ 3

2j + 3

2k 0.4643 0.5200 2.1538 3.8462 0.49

17 5

2+ 5

2i+ 3

2j + 3

2k 0.3542 0.3864 2.8235 5.1765 0.38

25 9

2+ 3

2i+ 3

2j + 1

2k 0.2404 0.2551 4.1600 7.8400 0.26

29 9

2+ 5

2i+ 3

2j + 1

2k 0.2071 0.2180 4.8276 9.1724 0.22

37 11

2+ 5

2i+ 1

2j+ 1

2k 0.1623 0.1690 6.1622 11.8378 0.18

41 11

2+ 5

2i+ 3

2j+ 3

2k 0.1464 0.1519 6.8293 13.1707 0.16

53 13

2+ 5

2i+ 3

2j+ 3

2k 0.1132 0.1165 8.8302 17.1698 0.12

61 15

2+ 3

2i+ 3

2j+ 1

2k 0.0984 0.1008 10.1639 19.8361 0.10

65 11

2+ 11

2i+ 3

2j+ 3

2k 0.0923 0.0945 10.8308 21.1692 0.10

73 17

2+ 1

2i+ 1

2j+ 1

2k 0.0822 0.0839 12.1644 23.8356 0.09

85 13

2+ 13

2i+ 1

2j+ 1

2k 0.0706 0.0719 14.1467 27.8353 0.08

89 17

2+ 7

2i+ 3

2j+ 3

2k 0.0674 0.0686 14.8315 29.1685 0.08

97 19

2+ 5

2i+ 1

2j+ 1

2k 0.0619 0.0628 16.1649 31.8351 0.06

Example 4.2. In Table II, we present the performance of the proposed Hur-witz constellation constructed by encoder Hurwitz integers whose each com-ponent is in Z + 1

2 over the AWGN channel by means of average energy,CFM, and SNR coding gains. The proposed Hurwitz constellations in TableII have advantage performances for transmission over the AWGN channel byset partitioning property.

There also exist different proposed primitive Hurwitz integers used toconstruct proposed Hurwitz constellations that have higher CFM and loweraverage energy in equal size. You can examine the following examples. Thebelow examples are given clues about the construction of tables.

Example 4.3. We consider proposed Hurwitz constellation(s) with N = 3 ·13 = 39 elements. There exist four different proposed primitive Hurwitz inte-gers used to construct proposed Hurwitz constellations with N = 39. These


Table 2. Table of average energies, CFMs and SNR coding gainsof proposed Hurwitz constellation constructed by proposed Hurwitz

integers whose coefficients are in Z + 1

2(N : The number of elements

(norm) of a proposed Hurwitz signal constellation, c: The number ofsubsets of a proposed Hurwitz signal constellation, d:The number of

elements (norm) of subset of a proposed Hurwitz signal constellation )

Constellations

CFM Energy

Proposed SNR

N c d Primitive Gauss Hurwitz Proposed Proposed gain

Hurwitz Hurwitz Hurwitz [dB]

Integers

15 3 5 5

2+ 5

2i+ 3

2j + 1

2k 1.2500 1.6667 1.2500 4.8000 -1.25

39 3 13 9

2+ 7

2i+ 5

2j + 1

2k 0.4643 0.5200 1.5600 11.5385 4.77

51 3 17 11

2+ 7

2i+ 5

2j + 3

2k 0.3542 0.3864 1.1591 15.5294 4.77

75 3 25 13

2+ 9

2i+ 7

2j + 1

2k 0.2404 0.2551 0.7653 23.5200 4.77

87 3 29 13

2+ 11

2i+ 7

2j+ 3

2k 0.2071 0.2180 0.6541 27.5172 4.77

185 5 37 21

2+ 13

2i+ 9

2j+ 7

2k 0.1673 0.1690 0.8447 59.1982 6.99

205 5 41 19

2+ 17

2i+ 11

2j+ 7

2k 0.1464 0.1519 0.7593 65.8537 6.99

265 5 53 23

2+ 19

2i+ 13

2j+ 1

2k 0.1132 0.1165 0.5824 85.8491 6.99

427 7 61 33

2+ 21

2i+ 13

2j+ 3

2k 0.0984 0.1008 0.7058 138.8520 8.45

455 7 65 27

2+ 25

2i+ 21

2j+ 5

2k 0.0923 0.0945 0.4717 148.4000 6.98

511 7 73 33

2+ 21

2i+ 17

2j+ 15

2k 0.0822 0.0839 0.5874 166.8490 8.45

595 7 85 37

2+ 29

2i+ 13

2j+ 1

2k 0.0706 0.0719 0.5030 194.8470 8.45

623 7 89 35

2+ 33

2i+ 13

2j+ 3

2k 0.0674 0.0686 0.4800 204.1800 8.45

873 9 97 41

2+ 35

2i+ 19

2j+ 15

2k 0.0619 0.0628 0.5654 286.5150 9.54

proposed primitive Hurwitz integers are 72 +

72 i+

72j +

32k,

92 +

52 i+

52j +

52k,

92 + 7

2 i +52j + 1

2k, and112 + 5

2 i +32j + 1

2k. There is no Gaussian constel-lation that has an equal size with the proposed Hurwitz constellation thathas N = 39 elements. Note that proposed primitive Hurwitz integers arenot to be the same size as primitive Gaussian integers to apply set parti-tioning property for proposed primitive Hurwitz integers. Firstly, we con-sider 9

2 + 72 i +

52j +

12k, and

112 + 5

2 i +32j +

12k proposed primitive Hurwitz


integers. For H 92+

72 i+

52 j+

12k

proposed Hurwitz constellation, the minimum

squared Euclidean distance, CFM and average energy are 1, 12.5128 and0.1598, respectively. The H 9

2+72 i+

52 j+

12k

proposed Hurwitz constellation is

partition the c = 3 different subsets with each set d = 13 elements. So, theSNR coding gain of H 9

2+72 i+

52 j+

12k

proposed Hurwitz constellation is com-

puted by using a subset of H 92+

72 i+

52 j+

12k

proposed Hurwitz constellation.

This subset is the set H(0)92+

72 i+

52 j+

12k

with 13 elements. Because, the minimum

squared Euclidean distance of H(0)92+

72 i+

52 j+

12k

constellation is larger than the

minimum squared Euclidean distance of H(1)92+

72 i+

52 j+

12k

and H(2)92+

72 i+

52 j+

12k

constellations. The minimum squared Euclidean distance, CFM and aver-

age energy of H(0)92+

72 i+

52 j+

12k

Hurwitz signal constellation that is a subset of

H 92+

72 i+

52 j+

12k

Hurwitz constellation are 9, 11.5385 and 1.5600, respectively.

Also, the primitive Hurwitz integers used to construct the Hurwitz constella-

tion that is the equivalence H(0)92+

72 i+

52 j+

12k

constellation with 13 elements are52+

32 i+

32j+

32k,

52+

52 i+

12j+

12k, and

72+

12 i+

12j+

12k. The minimum squared

Euclidean distance, CFM and average energy of these Hurwitz constellationsis 1, 0.5200, and 3.8462. The minimum squared Euclidean distance, CFMand average energy of the Gaussian constellation G3+2i with 13 elements are1, 2.1539 and 0.4643, respectively. We consider H 5

2+32 i+

32 j+

32k

primitive Hur-

witz integer calculating SNR coding gain. Therefore, SNR coding gain ofH 9

2+72 i+

52 j+

12k

proposed Hurwitz constellation is

SNR = −10 log(CFM of H

(0)92+

72 i+

52 j+

12k

CFM of H 52+

32 i+

32 j+

32k

) = −10 log(1.5600

0.5200) = 4.77. (4.9)

For H 112 + 5

2 i+32 j+

12k

proposed Hurwitz constellation, the minimum squared

Euclidean distance, CFM and average energy are 1, 12.5128 and 0.1598, re-spectively. The H 11

2 + 52 i+

32 j+

12k

proposed Hurwitz constellation is partition

the c = 3 different subsets with each set d = 13 elements. So, the SNR cod-ing gain of H 11

2 + 52 i+

32 j+

12k

proposed Hurwitz constellation is computed by

using a subset of H 112 + 5

2 i+32 j+

12k

proposed Hurwitz constellation. This subset

is the set H(0)112 + 5

2 i+32 j+

12k

with 13 elements. Because, the minimum squared

Euclidean distance ofH(0)112 + 5

2 i+32 j+

12k

is larger than the minimum squared Eu-

clidean distance of L(1)112 + 5

2 i+32 j+

12k

andH(2)112 + 5

2 i+32 j+

12k. The minimum squared

Euclidean distance, CFM and average energy of H(0)112 + 5

2 i+32 j+

12k

Hurwitz sig-

nal constellation that is a subset of H 112 + 5

2 i+32 j+

12k

proposed Hurwitz constel-

lation are 9, 11.5385 and 1.5600, respectively. The average energy, CFM andminimum squared Euclidean distance of H 9

2+72 i+

52 j+

12k

and H 112 + 5

2 i+32 j+

12k

proposed Hurwitz constellations have the same. So,

H 92+

72 i+

52 j+

12k

and H 112 + 5

2 i+32 j+

12k


proposed Hurwitz constellations have the same performances for transmis-sion over AWGN channel. Lastly, we consider 7

2 + 72 i +

72j + 3

2k, and92 +

52 i +

52j +

52k proposed primitive Hurwitz integers. For both constellations,

the minimum squared Euclidean distance, CFM and average energy are 1,12.5128 and 0.1598. The subsets of these proposed Hurwitz constellations are

H(0)72+

72 i+

72 j+

32k,

and H(0)92+

52 i+

52 j+

52k. The average energy, CFM and minimum

squared Euclidean distance of these subsets are 3, 11.5385 and 0.5200, respec-tively. Therefore, SNR coding gain of these proposed Hurwitz constellationsis


(0)72+

72 i+

72 j+

32k

CFM of H 52+

32 i+

32 j+

32k

) = −10 log(0.5200

0.5200) = 0. (4.10)

Consequently, the constellations that have higher CFM and larger minimumsquare Euclidean distance are H 9

2+72 i+

52 j+

12k

and H 112 + 5

2 i+32 j+

12k

proposed

Hurwitz constellations. We choose 52+

32 i+

32j+

32k proposed primitive Hurwitz

integer to represent in Table I, and 92 +

72 i+

52j+

12k primitive Hurwitz integer

to represent in Table II to create regular tables that are not crowded.

Example 4.4. We consider proposed Hurwitz constellation(s) with N = 3 ·29 = 87 elements. There exist eight different proposed primitive Hurwitz in-tegers used to construct proposed Hurwitz constellations with N = 87. Theseproposed primitive Hurwitz integers are 11

2 + 112 i+ 9

2j+52k,

132 + 9

2 i+72j+

72k,

132 + 11

2 i+ 72j+

32k,

132 + 13

2 i+ 32j+

12k,

152 + 7

2 i+72j+

52k,

152 + 11

2 i+ 12j+

12k,

172 + 5

2 i +52j +

32k, and

172 + 7

2 i +32j +

12k. There is no Gaussian constella-

tion that has an equal size with the proposed Hurwitz constellation that hasN = 87 elements. The minimum squared Euclidean distance, CFM and aver-age energy of proposed Hurwitz constellations constructed by these proposedprimitive Hurwitz integers are 1, 28.5057 and 0.0702, respectively. These pro-posed Hurwitz constellations are partition the c = 3 different subsets witheach set d = 29 elements. So, the SNR coding gains of these proposed Hurwitzconstellation is separately computed by using a subset of these proposed Hur-

witz constellations. We considerH(0)132 + 11

2 i+ 72 j+

32k, andH

(0)172 + 7

2 i+532 j+ 1

2ksubsets

with 29 elements of H 132 + 11

2 i+ 72 j+

32k, and H 17

2 + 72 i+

32 j+

12k, respectively. The

minimum square Euclidean distance of these subsets is larger than others.The minimum square Euclidean distance of these subsets is 9 but the othersis 6. Also average energy and CFM of these subsets are 27.5172 and 0.6541but the others are 27.5172 and 0.4361, respectively. Also, the primitive Hur-witz integers used to construct the Hurwitz constellations that is the equiv-

alence H(0)132 + 11

2 i+ 72 j+

32k, and H

(0)172 + 7

2 i+532 j+ 1

2kconstellations with 29 elements

are 72+

72 i+

32j+

32k, and

92+

52 i+

32j+

12k. The minimum squared Euclidean dis-

tance, CFM and average energy of H 72+

52 i+

52 j+

12k

Hurwitz constellation are 1,

0.2525, and 7.9200. The minimum squared Euclidean distance, CFM and av-erage energy of H 7

2+72 i+

32 j+

32k, and H 9

2+52 i+

32 j+

12k

Hurwitz constellations are


1, 0.2180, and 9.1724. We consider H 92+

52 i+

32 j+

12k

and 132 + 11

2 i+ 72j+

32k prim-

itive Hurwitz integers calculating SNR coding gain. The minimum squaredEuclidean distance, CFM and average energy of the Gaussian constellationG5+2i with 29 elements are 1, 4.8276 and 0.2071, respectively. Therefore, SNRcoding gain of H 13

2 + 112 i+ 7

2 j+32k

and proposed Hurwitz constellation is


(0)132 + 11

2 i+ 72 j+

32k

CFM of H 92+

52 i+

32 j+

312 k

) = −10 log(0.6541

0.2180) = 4.77.

(4.11)Consequently, the constellations that have higher CFM and larger minimumsquare Euclidean distance are H 13

2 + 112 i+ 7

2 j+32k

and H 172 + 7

2 i+32 j+

12k

proposed

Hurwitz constellations. We choose 132 + 11

2 i + 72j + 3

2k proposed primitive

Hurwitz integer to represent in Table I, and 92 + 5

2 i +32j + 1

2k primitiveHurwitz integer to represent in Table II to create regular tables that are notcrowded.

5. Conclusion

In this study, we investigated Hurwitz integers that have ”division with smallremainder” property and defined a new set, which is formed Hurwitz integersthat have ”division with small remainder” property, named encoder Hur-witz integers. So, we can define a Euclidean metric for Hurwitz constellationsthat lies on the Hurwitz integers or, different appropriate metrics for codesover the rings of Hurwitz integers. We showed that the Euclid division al-gorithm whether works or not for Hurwitz integers whose coefficients are inZ with proposition 3.1 and proposition 3.2. Whichever technique we use, wecan check whether the Euclidean division algorithm works for Hurwitz in-tegers with these propositions (see example 3.5). In addition, we examinedthe performances of the Hurwitz integers whose components are in Z+ 1

2 oftransmission over the AWGN channel. Also, the modulo function defined inthis paper shows an inappropriate technique for Hurwitz constellations. Newtechniques can improvement such as Rohweder et al.[19], and Guzeltepe [18].In our forward study, we will be investigated the performances of Hurwitzconstellations for transmission over the AWGN channel by a new techniquesuch as technique in [18]. Therefore, this paper is written to be a referencein our following study and colleagues’ forward studies.

References

[1] Ian F. Blake, ”Codes over certain rings,” Information and Control, Volume 20,Issue 4, Pages 396-404, May 1972.

[2] Ian F. Blake, ”Codes over integer residue rings,” Information and Control, Vol-ume 29, Issue 4, Pages 295-300, December 1975.

[3] Eugene Spiegel, ”Codes over Zm,” Information and Control, Volume 35, Issue1, Pages 48-51, September 1977.


[4] Eugene Spiegel, ”Codes over Zm(revisited),” Information and Control, Volume37, Issue 1, Pages 100-104, April 1978.

[5] Chandra Satyanarayana, ”Lee metric codes over integer residue rings (Cor-resp.),” IEEE Transactions on Information Theory, Volume 25, Issue 2, March1979.

[6] A. R. Calderbank and N. J. A. Sloane, ”Modular and p-adic cyclic codes,”Designs, Codes and Cryptography, Volume 6, Issue 1, Pages 21-35, July 1995.

[7] Trajano Pires da Nobrega Neto, J. Carmelo Interlando, Osvaldo Milare Favareto,Michele Elia and Reginaldo Palazzo, ”Lattice constellations and codes fromquadratic number fields,” IEEE Transactions on Information Theory, Volume47, Issue 4, Pages 1514-1527, May 2001.

[8] K. Abdelmoumen, H. Ben-azza and M. Najmeddine, ”About Euclidean codes inrings,” British Journal of Mathematics and Computer Science, Volume 4, Issue10, Pages 1356-1364, 16-31 May 2014.

[9] Klaus Huber, ”Codes over Gaussian integers,” IEEE Transactions on Informa-tion Theory, Volume 40, Issue 1, pp 207-216, January 1994.

[10] Stefka Bouyuklieva, ”Applications of the Gaussian integers in coding theory,”Proceedings of the 3rd International Colloquium on Differential Geometry andits Related Fields, September 3-7,2012.

[11] Jurgen Freudenberger, Farhad Ghaboussi and Sergo Shavgulidze, ”New codingtechniques for codes over Gaussian integers,” IEEE Transactions on Communi-cations, Volume 61, Issue 8, Pages 3114-3124, August 2013.

[12] Jurgen Freudenberger, Farhad Ghaboussi and Sergo Shavgulidze, ”Set Parti-tioning and Multilevel Coding for Codes Over Gaussian Integer Rings,” SCC2013; 9th International ITG Conference on Systems, Communication and Cod-ing, Munchen, Deutschland, Pages 1-5, 21-24 January 2013.

[13] Mehmet Ozen and Murat Guzeltepe, ”Cyclic codes over some finite rings,”Selcuk Journal of Applied Mathematics, Volume 11, No. 2, Pages 71-76, 2010.

[14] Mehmet Ozen and Murat Guzeltepe, ”Codes over quaternion integers,” Euro-pean Journal of Pure and Applied Mathematics, Volume 3, No. 4, Pages 670-677,July 2010.

[15] Young Ju Choie and Steven T. Dougherty, ”Codes over Z2m and Jacobi formsover the quaternions,” Applicable Algebra in Engineering, Communication andComputing, Volume 15, Issue 2, Pages 129-147, September 2004.

[16] Mehmet Ozen and Murat Guzeltepe, ”Cyclic codes over some finite quaternioninteger rings”, Journal of the Franklin Institute, Volume 348, Issue 7, Pages1312-1317, September 2011.

[17] Tariq Shah and Summera Said Rasool, ”On codes over quaternion integers,”Applicable Algebra in Engineering, Communication and Computing, Volume24, Issue 6, December 2013.

[18] Murat Guzeltepe, ”Codes over Hurwitz integers,” Discrete Mathematics , Vol-ume 313, Issue 5, Pages 704-714, March 2013.

[19] D. Rohweder, S. Stern, R. F. H. Fischer, S. Shavgulidze and J. Freuden-berger, ”Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, De-tection, and Multilevel Coding,” in IEEE Transactions on Communications, doi:10.1109/TCOMM.2021.3083323.


[20] Murat Guzeltepe, ”On some perfect codes over Hurwitz integers,” Mathemat-ical Advances in Pure and Applied Sciences, Volume 1, No. 1, Page 39-45, May18, 2018.

[21] Murat Guzeltepe and Alev Altınel, ”Perfect 1-error-correcting Hurwitz weightcodes,” Mathematical Communications, Volume 22, No. 2, Pages 265-272, 2017.

[22] Murat Guzeltepe and Olof Heden, ”Perfect Mannheim, Lipschitz and Hurwitzweight codes,” Mathematical Communications, Volume 19, No. 2, Pages 253-276, 2014.

[23] Giuliana Davidoff, Peter Sarnak and Alain Valette, ”Elementary Number The-ory, Group Theory, and Ramanujan Graphs,” Cambridge University Press, 2003.

[24] John Horton Conway and Derek Alan Smith, ”On Quaternions and Octonions:Their Geometry, Arithmetic, and Symmetry,” Natick, MA, USA: A.K. PetersLtd., 2003.

[25] Jurgen Freudenberger and Sergo Shavgulidze, ”New four-dimensional signalconstellations from Lipschitz integers for transmission over the Gaussian chan-nel,” IEEE Transactions on Communications, Volume 63, Issue 7, Pages 2420-2427, July 2015.

[26] G. Forney and L.-F. Wei, ”Multidimensional constellations. I. Introduc-tion,figures of merit, and generalized cross constellations,” IEEE Journal on SelectedAreas in Communications, Volume 7, Number 6, Pages 877–892, August 1989.

Ramazan DuranDepartment Mathematics,Faculty of Arts and Sciences,TR 03200 Afyonkarahisar,Turkeye-mail: [email protected] Mathematics,Faculty of Arts and Sciences,TR 54187 Sakarya,Turkeye-mail: [email protected]

Murat GuzeltepeDepartment Mathematics,Faculty of Arts and Sciences,TR 54187 Sakarya,Turkeye-mail: [email protected]

Encoder Hurwitz Integers: The Hurwitz integers that have ...

Documents

Transcript of Encoder Hurwitz Integers: The Hurwitz integers that have ...