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Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: In this article, the Gray image of 1+2u+2v+2uv was used to obtain cyclic, quasi-cyclic and permutation equivalent codes over the ring of integers modulo 4.
Abstract: Let $${\mathbb {Z}}_{4}$$
be the ring of integers modulo 4. This paper presents $$(1+2u+2v+2uv)$$
-constacyclic and skew $$(1+2u+2v+2uv)$$
-constacyclic codes over the ring $$ {\mathbb {Z}}_{4} +u{\mathbb {Z}}_{4}+v{\mathbb {Z}}_{4}+uv{\mathbb {Z}}_{4} $$
where $$u^2=u,v^{2}=v, uv=vu$$
. We define three Gray maps and show that the Gray images of $$(1+2u+2v+2uv)$$
-constacyclic and skew $$(1+2u+2v+2uv)$$
-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over $${\mathbb {Z}}_4$$
. Also, we show that cyclic and $$(1+2u+2v+2uv)$$
-constacyclic codes of odd length are principally generated. As an application, several new quaternary linear codes from the Gray images of $$(1+2u+2v+2uv)$$
-constacyclic codes are obtained.
3 citations
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TL;DR: A generalization of the well-known Euler's theorem is derived which can be used to determine all the generators of a given NTT once the generators in the underlying finite field are identified.
Abstract: In this paper, results are presented that can be used to obtain all the possible generators for a number theoretic transform (NTT) defined in a finite integer ring and its polynomial extensions. A generalization of the well-known Euler's theorem is derived which can be used to determine all the generators of a given NTT once the generators in the underlying finite field are identified. Based on this extension, a procedure is also described to compute cyclotomic factorization in these rings. This factorization and Chinese remainder theorem lead to computationally efficient algorithms for computing cyclic convolution of two sequences defined in finite and complex integer rings.
3 citations
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TL;DR: It is shown that x e N < iyiz[x =y2 V (y2 1 +XZ2 A y3 0 y)] and that it is impossible to define N using a formula with a single quantifier.
Abstract: We give necessary conditions for a set to be definable by a formula with a universal quantifier and an existential quantifier over algebraic integer rings or algebraic number fields. From these necessary conditions we obtain some undefinability results. For example, N is not definable by such a formula over Z. This extends a previous result of R. M. Robinson. Introduction. We know very little about arithmetic definability even for formulas with a single quantifer. For instance, let f (x, y) be a polynomial over Z and let Af c Z be defined as follows: x Af Z ly Jf(x, y) = 0. For an arbitrary polynomial f we cannot tell whether Af is nonempty or not, because Af is nonempty if and only if f (x, y) = 0 is solvable in Z, but there is no known algorithm to decide the solvability of Diophantine equations in two unknowns. In his paper [2], R. M. Robinson is concerned with the following question: What is the simplest possible arithmetical definition of the set of natural numbers N in the ring of integers Z? In particular, what is the smallest number of quantifiers which can occur in such a definition? He showed that x e N < iyiz[x =y2 V (y2 1 +XZ2 A y3 0 y)]. He also showed that it is impossible to define N using a formula with a single quantifier. If we consider this problem just by counting the number of quantifiers, then these certainly are the best possible results. However, we can ask further: What kind of quantifiers are necessary in the defining formula? Then it is possible to obtain better results. Let Q denote the quantifier V or ]. We say p(x) is a Q1 Q2 formula if and only if (p(x) is logically equivalent to a formula of the form Q1yQ2z0(x, y, z), where +(x, y, z) is a formula containing no quantifiers and no free variables except x, y and z. The set Received April 19, 1990; revised December 17, 1990. 1991 Mathematics Subject Classification. Primary 03C40.
3 citations
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TL;DR: In this paper, the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over a real number field was applied to study the growth of the invariants of the ring of integers of a positive integer.
Abstract: We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the {\em $g$-invariants} of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal O$ be the ring of integers of $K$ and $g_{\mathcal O}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal O$-linear forms must be a sum of $g_{\mathcal O}(n)$ squares of $n$-ary $\mathcal O$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\mathcal O}(n)$ is at most an exponential of $\sqrt{n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\mathcal O}(n)$ for rings of integers $\mathcal O$ other than $\mathbb Z$.
3 citations
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TL;DR: In this paper, a modified quadratic form on a family of Euclidean lattices that arise from any CM number field is defined and a bound on the field norm of any vector that has a minimal length in any of these lattices, in terms of a basis for the group of units of the ring of integers of the field.
Abstract: This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that contains the standard field trace as a special case. Using this modified quadratic form, we obtain a bound on the field norm of any vector that has a minimal length in any of these lattices, in terms of a basis for the group of units of the ring of integers of the field. For any CM number field F, we prove that there exists a finite set of elements of F which allows one to find the set of minimal vectors in every principal ideal of the ring of integers of F. We interpret our result in terms of the asymptotic behavior of a Hilbert modular form, and consider some of the computational implications of our theorem. Additionally, we show how our result can be applied to the specific Craig's Difference Lattice problem, which asks us to find the minimal vectors in lattices arising from cyclotomic number fields.
3 citations