Residue Class Ring Research Articles

Synthesis of Self-Checking Circuits for Train Route Traffic Control at Intermediate Stations with Control of Calculations Based on Weight-Based Sum Codes

When synthesizing systems for railway interlocking, it is recommended to use automated models to implement the logic of railway automation and remote control units. Finite-state machines (FSMs) can be implemented on any hardware component. When using relay technology, the functional safety of electrical interlocking is achieved by using uncontrolled (safety) relays with a high coefficient of asymmetry of failures in types 1 → 0 and 0 → 1. When using programmable components, the use of backup and diverse protection methods is required. This paper presents a flexible approach to synthesizing FSMs for railway automation and remote control units that offer both individual and route-based control. Unlike existing solutions, this proposal considers the pre-failure states of railway automation and remote control units during the finite-state machine synthesis stage. This enables the implementation of self-checking and self-diagnostic modules to manage automation units. By increasing the number of states for individual devices and considering the states of interconnected objects, the transition graphs can be expanded. This expansion allows for the synthesis of the transition graph of the control subsystem and other systems. The authors used a field-programmable gate array (FPGA) to implement a finite-state machine. In this case, the proposal is to encode the states of a finite-state machine using weight-based sum codes in the residue class ring based on a given modulus. The best coverage of errors occurring at the outputs of the logic converter in the structure of the FSM can be ensured by selecting the weighting coefficients and the value of the module. This paper presents an example of synthesizing an FPGA-based FSM using state encoding through modular weight-based sum codes. The operation of the synthesized device was modeled. It was found to operate according to the same algorithm as the real devices. When synthesizing self-checking and self-controlled train control devices, it is recommended to consider the solutions proposed in this paper.

Some Properties of Subspaces over Residue Class Rings

Some Properties of Subspaces over Residue Class Rings

Subspace Sum Graphs over Residue Class Rings

Subspace Sum Graphs over Residue Class Rings

A zero-sum problem related to the max gap of the unit group of the residue class ring

Let S be a sequence over a finite abelian group G and vg(S) be the times that g∈G occurs in S. A sequence S over G is called weak-regular if vg(S)≤ord(g) for every g∈G. Denote by N(G) the smallest integer t such that every weak-regular sequence S over G of length |S|≥t has a nonempty zero-sum subsequence T of S satisfying vg(T)=vg(S) for some g|S. N(G) has been formulated by Gao et al. very recently to study zero-sum problems in a unify way and determined only for cyclic groups of prime-power order and some other very special groups. As for general cyclic groups G=Cn, they gave that2n−⌈3n⌉+1≤N(G)≤2n−⌈2n+1⌉+1.In this paper, we first study the max gap of the unit group of the residue class ring and give an upper bound of it. Then we prove that there is always an integer a∈[n12,n12+n14] such that gcd⁡(a,n)=1 for n≥2227. Finally, we improve the result of Gao et al. by showing that2n−⌈2n+1⌉≤N(G)≤2n−⌈2n+1⌉+1 for any cyclic group G=Cn with n≥3, in which for each equality, there are infinitely many n making it hold. And a computing result prefigures that N(G) has not been determined only for very few cyclic groups G.

Implicit linear difference equation over residue class rings

We investigate the first order implicit linear difference equation over residue class rings modulo m. We prove an existence criterion and establish the number of solutions for this equation. We obtain analogous results for the initial problem of the considered equation. The examples which illustrate the developed theory are given.

Edge Domination of the Nilpotent Cayley Graph of the Residue Class Ring

In their earlier study, the authors have obtained the properties of the nilpotent Cayley graph G (Zn,N) associated with the set of nilpotent elements N of the residue class ring (Zn, ⊕, ), modulo an integer n ≥ 1 and its vertex domination parameters. In this paper, the edge cover and its associated parameters of this graph are found.

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

Erdős–Ko–Rado theorem for vector spaces over residue class rings

Let [Formula: see text] be its decomposition into a product of powers of distinct primes, and [Formula: see text] be the residue class ring modulo [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-dimensional row vector space over [Formula: see text]. A generalized Grassmann graph over [Formula: see text], denoted by [Formula: see text] ([Formula: see text] for short), has all [Formula: see text]-subspaces of [Formula: see text] as its vertices, and two distinct vertices are adjacent if their intersection is of dimension [Formula: see text], where [Formula: see text]. In this paper, we determine the clique number and geometric structures of maximum cliques of [Formula: see text]. As a result, we obtain the Erdős–Ko–Rado theorem for [Formula: see text].

On the structure of sets in a residue class ring with the same representation function

Let m be an integer with m≥2. For a given set S⊆Zm and n‾∈Zm, let RS(n‾) denote the number of solutions of the equation n‾=s‾+s′‾ with unordered pair (s‾,s′‾)∈S×S and s‾≠s′‾. In this paper, we determine the structure of sets A,B satisfying A∪B=Zm, |A∩B|=2 and RA(n‾)=RB(n‾) for all n‾∈Zm, where m is even.

Some algebraic Invariants of the residue class rings of the edge ideals of perfect semiregular trees

Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.

The wiener index of the zero-divisor graph for a new class of residue class rings

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo p s q t for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph Γ E (R) are derived for .

A congruence equation involving the factorisation in residue class ring mod $n$

A congruence equation involving the factorisation in residue class ring mod $n$

On the sums of squares of exceptional units in residue class rings

Let \(n\ge 1, e\ge 1, k\ge 2\) and c be integers. An integer u is called a unit in the ring \({\mathbb {Z}}_n\) of residue classes modulo n if \(\gcd (u, n)=1\). A unit u is called an exceptional unit in the ring \({\mathbb {Z}}_n\) if \(\gcd (1-u,n)=1\). We denote by \({\mathcal {N}}_{k,c,e}(n)\) the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^e+\cdots +x_k^e\equiv c \pmod n\) with all \(x_i\) being exceptional units in the ring \({\mathbb {Z}}_n\). In 2017, Mollahajiaghaei presented a formula for the number of solutions \((x_1,\ldots ,x_k)\) of the congruence \(x_1^2+\cdots +x_k^2\equiv c\pmod n\) with all \(x_i\) being the units in the ring \({\mathbb {Z}}_n\). Meanwhile, Yang and Zhao gave an exact formula for \({\mathcal {N}}_{k,c,1}(n)\). In this paper, by using Hensel’s lemma, exponential sums and quadratic Gauss sums, we derive an explicit formula for the number \({\mathcal {N}}_{k,c,2}(n)\). Our result extends Mollahajiaghaei’s theorem and that of Yang and Zhao.

A Paradigmatic Approach to Find Equal Sum Partitions of Zero-Divisors via Complete Graphs

In computer science and mathematics, a partition of a set into two or more disjoint subsets with equal sums is a well-known NP-complete problem. This is a hard problem and referred to as the partition problem or number partitioning. In this paper, we solve a particular type of NP-complete problem on the set of all zero-divisors of Z n including zero, where Z n is the ring of residue classes of a positive integer n . In this regard, we introduce and investigate quadratic zero-divisor graph in which we build an edge between zero-divisors z i and z j if and only if z i 2 ≡ z j 2 mod n , i ≠ j . This is denoted as G ⏞ 2 , n . We characterize these graphs in term of complete graphs for classes of integers 2 α , p α , 2 α p , 2 p α and p q , where α is any positive integer and p , q are odd primes.

Polynomial functions on rings of dual numbers over residue class rings of the integers

AbstractThe ring of dual numbers over a ringRisR[α] =R[x]/(x2), whereαdenotesx+ (x2). For any finite commutative ringR, we characterize null polynomials and permutation polynomials onR[α] in terms of the functions induced by their coordinate polynomials (f1,f2∈R[x], wheref=f1+αf2) and their formal derivatives onR.We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤpn[α] forn≤p(pprime).

Systems of parameters and the Cohen–Macaulay property

We recall numerical criteria for Cohen–Macaulayness related to system of parameters and introduce monomial ideals of König type which include the edge ideals of König graphs. We show that a monomial ideal is of König type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form \(x_i-x_j\). This provides an algebraic characterization of König graphs. We use this special parameter systems for the study of the edge ideal of König graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex \(\Delta \) we introduce a system of parameters for \(K[\Delta ]\) with a universal construction principle, independent of the base field and only dependent on the faces of \(\Delta \). This system of parameters is an efficient tool to test Cohen–Macaulayness of the Stanley–Reisner ring of a simplicial complex.

New Generalized Cyclotomic Quaternary Sequences with Large Linear Complexity and a Product of Two Primes Period

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

A novel approach to find partitions of $ Z_{m} $ with equal sum subsets via complete graphs

<abstract> In mathematics and computer sciences, the partitioning of a set into two or more disjoint subsets of equal sums is a well-known NP-complete problem, also referred to as partition problem. There are various approaches to overcome this problem for some particular choice of integers. Here, we use quadratic residue graph to determine the possible partitions of positive integers $ m = 2^{\beta}, q^{\beta}, 2^{\beta}q, $ $ 2q^{\beta}, qp, $ where $ p $, $ q $ are odd primes and $ \beta $ is any positive integer. The quadratic residue graph is defined on the set $ Z_{m} = \{\overline{0}, \overline{1}, \cdots, \overline{m-1}\}, $ where $ Z_{m} $ is the ring of residue classes of $ m $, i.e., there is an edge between $ \overline{x}, $ $ \overline{y}\in Z_{m} $ if and only if $ \overline{x}^{2}\equiv \overline{y}^{2}\; (\text{mod}\; m) $. We characterize these graphs in terms of complete graph for some particular classes of $ m $. </abstract>

Endomorphisms of semicyclic n-groups

are isomorphic to (𝑛, 2)-nearrings of endomorphisms of certain semiabelian 𝑛-groups. Such almost (𝑛, 2)-nearrings are found for semicyclic 𝑛-groups. On the additive group of integers 𝑍 we construct an abelian 𝑛-group ⟨𝑍, 𝑓1⟩ with 𝑛- ary operation 𝑓1(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 𝑙, where 𝑙 is any integer. For a nonidentical automorphism 𝜙(𝑧) = −𝑧 on 𝑍, we can specify semiabelian 𝑛-group ⟨𝑍, 𝑓2⟩ for 𝑛 = 2𝑘 + 1, 𝑘 ∈ 𝑁, with the 𝑛-ary operation 𝑓2(𝑧1, . . . , 𝑧𝑛) = 𝑧1 − 𝑧2 + . . . + 𝑧2𝑘−1 − 𝑧2𝑘 + 𝑧2𝑘+1. Any infinite semicyclic 𝑛-group is isomorphic to either the 𝑛-group ⟨𝑍, 𝑓1⟩, where 0 ≤ 𝑙 ≤ [𝑛−1 2 ], or the 𝑛-group ⟨𝑍, 𝑓2⟩ for odd 𝑛. In the first case we will say that such 𝑛-group has type (∞, 1, 𝑙), and in the second case, it has type (∞,−1, 0). In 𝑍 we select the set 𝑃 = {𝑚|𝑚𝑙 ≡ 𝑙 (mod 𝑛 − 1)} and define an 𝑛-ary operation ℎ by the rule ℎ(𝑚1, . . . ,𝑚𝑛) = 𝑚1 + . . . + 𝑚𝑛 on this set. Then the algebra ⟨𝑃, ℎ, ·⟩, where · is the multiplication of integers, is a (𝑛, 2)-ring. It is proved that ⟨𝑃, ℎ, ·⟩ is isomorphic to (𝑛, 2)-ring of endomorphisms of semicyclic 𝑛-group of type (∞, 1, 𝑙). In the 𝑛-group ⟨𝑍 × 𝑍, ℎ⟩ = ⟨𝑍, 𝑓2⟩ × ⟨𝑍, 𝑓2⟩ we define the binary operation ◇ by the rule (𝑚1, 𝑢1) ◇ (𝑚2, 𝑢2) = (𝑚1𝑚2,𝑚1𝑢2 +𝑢1). Then ⟨𝑍 ×𝑍, ℎ, ◇⟩ is an (𝑛, 2)-nearringsg. It is proved that ⟨𝑍 ×𝑍, ℎ, ◇⟩ is isomorphic to (𝑛, 2)-nearrings of endomorphisms of a semicyclic 𝑛-group of type (∞,−1, 0). It is proved that (𝑛, 2)-ring ⟨𝑍, 𝑓, *⟩, where 𝑓(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 1 and 𝑧1 * 𝑧2 = 𝑧1𝑧2(𝑛 − 1) + 𝑧1 + 𝑧2, is isomorphic to (𝑛, 2)-rings of endomorphisms of infinite cyclic 𝑛-group. On additive group of the ring of residue classes of 𝑍𝑘 we define 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where the 𝑛- ary operation 𝑓3 operates according to the rule 𝑓3(𝑧1, . . . , 𝑧𝑛) = 𝑧1+𝑚𝑧2+. . .+𝑚𝑛−2𝑧𝑛−1+𝑧𝑛+𝑙, 1 ≤ 𝑚 < 𝑘 and 𝑚 is relatively prime to 𝑘. In addition, 𝑚 satisfies the congruence 𝑙𝑚 ≡ 𝑙 (mod 𝑘) and multiplicative order of 𝑚 modulo 𝑘 divides 𝑛 − 1. Any finite semicyclic 𝑛-group of order 𝑘 is isomorphic to 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where 𝑙 | gcd(𝑛 − 1, 𝑘) for 𝑚 = 1 and 𝑙 | gcd(𝑚𝑛−1−1 𝑚−1 , 𝑘) for 𝑚 = 1. We will say that such 𝑛-group has type (𝑘, 𝑚, 𝑙). In the 𝑛-group ⟨𝑃, ℎ⟩ = ⟨𝑍𝑘, 𝑓3⟩ × ⟨𝑍𝑙, 𝑓4⟩, 𝑓4(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + 𝑟𝑧2 + . . . + 𝑟𝑛−2𝑧𝑛−1 + 𝑧𝑛, where 𝑟 is the remainder of dividing 𝑚 by 𝑙, we define the binary operation ◇ by the rule $$(𝑢1, 𝑣1) ◇ (𝑢2, 𝑣2) = (𝑢2𝑠1 + 𝑢1, 𝑣2𝑠1 + 𝑣1)$$ where 𝑠1 ∈ 𝑍𝑘, (𝑠1 − 1 = 𝑠0 + 𝑣1)/𝑘 𝑙 , and 𝑠0 is solution of congruence 𝑥 ≡ (𝑛−1)𝑢1 𝑙 (mod 𝑘/𝑙 ) for 𝑚 = 1 and 𝑥 ≡ 𝑚𝑛−1−1 𝑚−1 𝑢1 𝑙 (mod 𝑘/𝑙 ) for 𝑚 = 1. It is proved that the algebra ⟨𝑃, ℎ, ◇⟩ is (𝑛, 2)-ring for 𝑚 = 1 and (𝑛, 2)-nearring for 𝑚 = 1, which is isomorphic to (𝑛, 2)-ring of endomorphisms of abelian semicyclic 𝑛-group of type (𝑘, 1, 𝑙) with 𝑚 = 1 and (𝑛, 2)-nearring of endomorphisms of semicyclic 𝑛-groups of type (𝑘, 𝑚, 𝑙) for 𝑚 = 1. It is proved that (𝑛, 2)-ring ⟨𝑍𝑘, 𝑓, *⟩, where 𝑓(𝑧1, 𝑙𝑑𝑜𝑡𝑠, 𝑧𝑛) = 𝑧1 + 𝑙𝑑𝑜𝑡𝑠 + 𝑧𝑛 + 1 and 𝑢1 *𝑢2 = 𝑢1 𝑐𝑑𝑜𝑡𝑢2 𝑐𝑑𝑜𝑡(𝑛−1)+𝑢1+𝑢2, is isomorphic to (𝑛, 2)-ring of endomorphisms of finite cyclic 𝑛-group of order 𝑘.

Three classes of partitioned difference families and their optimal constant composition codes

&lt;p style='text-indent:20px;'&gt;Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; prime or composite, cyclotomy on a residue class ring &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ {\mathbb{Z}}_{v} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [&lt;xref ref-type="bibr" rid="b40"&gt;40&lt;/xref&gt;], we introduce a generalized cyclotomy of order &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ e $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on the ring &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, where &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ q_i $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ q_j $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ i\neq j $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) may not be co-prime, which includes classical cyclotomy as a special case. Here, &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ q_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ q_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \cdots $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ q_k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are powers of primes with an integer &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ e|(q_i-1) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for any &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ 1\leq i\leq k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.&lt;/p&gt;