Codes over Muffin Ideals of Quaternion Integer Ring-(-1,-1/zp)

2019 ◽  
Vol 10 (1) ◽  
pp. 20-32
Author(s):  
Shaikh Javed Shafee ◽  
Arunkumar R. Patil
Keyword(s):  
Integer Ring

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

scholarly journals On the extended Kimʼs p -adic q -deformed fermionic integrals in the p -adic integer ring

2013 ◽  
Vol 133 (10) ◽  
pp. 3348-3361 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Erdoğan Şen
Keyword(s):  
Integer Ring

scholarly journals Quantum computing thanks to Bianchi groups

2019 ◽  
Vol 198 ◽  
pp. 00012 ◽  
Author(s):  
Michel Planat
Keyword(s):  
Quantum Computing ◽  
Trefoil Knot ◽  
Integer Ring ◽  
Universal Quantum ◽  
A Chain ◽  
Low Dimensional ◽  
Free Subgroups ◽  
Knots And Links ◽  
Good Agreement ◽  
Positive Operator Valued Measure

It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O10) (with O10 the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].

scholarly journals Explicit Formulas Involving -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong Jin Seo
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Derivative Operator ◽  
Euler Numbers And Polynomials ◽  
Integer Ring

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

scholarly journals A new twofold Cornacchia-type algorithm and its applications

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bei Wang ◽  
Yi Ouyang ◽  
Songsong Li ◽  
Honggang Hu
Keyword(s):  
Elliptic Curves ◽  
Characteristic Equation ◽  
Upper Bound ◽  
Hyperelliptic Curve ◽  
Gaussian Integer ◽  
Ring Of Integers ◽  
Euclidean Domain ◽  
Type Algorithm ◽  
Integer Ring ◽  
Unified Method

<p style='text-indent:20px;'>We focus on exploring more potential of Longa and Sica's algorithm (ASIACRYPT 2012), which is an elaborate iterated Cornacchia algorithm that can compute short bases for 4-GLV decompositions. The algorithm consists of two sub-algorithms, the first one in the ring of integers <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> and the second one in the Gaussian integer ring <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula>. We observe that <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula> in the second sub-algorithm can be replaced by another Euclidean domain <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ (\omega = \frac{-1+\sqrt{-3}}{2}) $\end{document}</tex-math></inline-formula>. As a consequence, we design a new twofold Cornacchia-type algorithm with a theoretic upper bound of output <inline-formula><tex-math id="M6">\begin{document}$ C\cdot n^{1/4} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ C = \frac{3+\sqrt{3}}{2}\sqrt{1+|r|+|s|} $\end{document}</tex-math></inline-formula> with small values <inline-formula><tex-math id="M8">\begin{document}$ r, s $\end{document}</tex-math></inline-formula> given by the curves.</p><p style='text-indent:20px;'>The new twofold algorithm can be used to compute <inline-formula><tex-math id="M9">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on two classes of curves. First it gives a new and unified method to compute all <inline-formula><tex-math id="M10">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on <inline-formula><tex-math id="M11">\begin{document}$ j $\end{document}</tex-math></inline-formula>-invariant <inline-formula><tex-math id="M12">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> elliptic curves over <inline-formula><tex-math id="M13">\begin{document}$ \mathbb{F}_{p^2} $\end{document}</tex-math></inline-formula>. Second it can be used to compute the <inline-formula><tex-math id="M14">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the Jacobian of the hyperelliptic curve defined as <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{C}/\mathbb{F}_{p}:y^{2} = x^{6}+ax^{3}+b $\end{document}</tex-math></inline-formula>, which has an endomorphism <inline-formula><tex-math id="M16">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> with the characteristic equation <inline-formula><tex-math id="M17">\begin{document}$ \phi^2+\phi+1 = 0 $\end{document}</tex-math></inline-formula> (hence <inline-formula><tex-math id="M18">\begin{document}$ \mathbb{Z}[\phi] = \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula>). As far as we know, none of the previous algorithms can be used to compute the <inline-formula><tex-math id="M19">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the latter class of curves.</p>

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