Finite Rings Research Articles

Counting points on diagonal equations over Galois rings GR(p2,2n)

Let R=GR(p2,2n) be a Galois ring and Fq a finite field, where q=p2n. In this paper, we study the number of solutions of equations of the form a1x1d1+⋯+asxsds=b with xi∈GR(p2,2ti), where ai,b∈R∖{0} and ti|n for all i=1,...,s, and employ results on quadratic forms over finite fields to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions.

Polyfunctions over commutative rings

A function [Formula: see text], where [Formula: see text] is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative [Formula: see text]. Based on this notion, we introduce ring invariants which associate to [Formula: see text] the numbers [Formula: see text] and [Formula: see text], where [Formula: see text] is the subring generated by [Formula: see text]. For the ring [Formula: see text] the invariant [Formula: see text] coincides with the number theoretic Smarandache or Kempner function [Formula: see text]. If every function in a ring [Formula: see text] is a polyfunction, then [Formula: see text] is a finite field according to the Rédei–Szele theorem, and it holds that [Formula: see text]. However, the condition [Formula: see text] does not imply that every function [Formula: see text] is a polyfunction. We classify all finite commutative rings [Formula: see text] with unit element which satisfy [Formula: see text]. For infinite rings [Formula: see text], we obtain a bound on the cardinality of the subring [Formula: see text] and for [Formula: see text] in terms of [Formula: see text]. In particular we show that [Formula: see text]. We also give two new proofs for the Rédei–Szele theorem which are based on our results.

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of FpFp[θ]-linear codes, where p∈{2;3} and θ≠θ2=0 that provides LCD codes over Fp.

Bounds for Coding Theory over Rings

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov’s weight over the integers modulo 2s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight.

Dynamics of magnetic flux propagation in bulk, single grain superconducting rings during pulsed field magnetisation

When used as trapped field magnets (TFMs), single grain, bulk high-temperature superconducting (HTS) rings are promising candidates for the generation of strong, uniform magnetic fields for nuclear magnetic resonance. The pulsed field magnetisation (PFM) technique provides a low cost, compact and portable method to magnetise these samples as TFMs; however it has proven difficult to achieve high trapped fields in HTS rings using PFM. To date, a record field of only 0.60 T has been achieved for rings magnetised by single-pulse PFM—compared with over 4 T for disc-shaped HTS—and the reasons for this discrepancy are poorly understood. In this work, we use the finite element method to model the propagation of magnetic flux into HTS rings under quasi-static zero field cooled magnetisation and PFM, and validate the results analytically and experimentally. Magnetic flux is found to penetrate finite HTS rings from both the inner and outer surfaces, inducing a negative field at the inner face of the ring. This field is reversed as the applied field increases past the point of full penetration, locally dissipating magnetic energy and heating the sample. HTS rings are therefore more susceptible to local instabilities that severely limit their ability to trap a useful magnetic field. Consequently, thermomagnetic stability of HTS rings during single-pulse PFM can only be ensured by taking careful consideration of reducing flux movement through the bulk around the point at which the field is reversed. This may require more advanced PFM techniques like waveform control or multi-pulse stepwise-cooling to reduce local heating and increase the trapped field.

LCP of group codes over finite Frobenius rings

LCP of group codes over finite Frobenius rings

$${{\,\mathrm{\texttt {VIC}}\,}}$$-modules over noncommutative rings

For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.

New Family of Polyphase Sequences with Low Correlation from Galois Rings

New Family of Polyphase Sequences with Low Correlation from Galois Rings

A Unique Representation of Cyclic Codes over GR(pn,r)

Let R be a Galois ring, GR(pn,r), of characteristic pn and of order pnr. In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, ps, where s=vp(N) and vp are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when n=2.

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree 2 n -1-n of the polynomial algebra 𝔽 2 [x 1 ,...,x n ], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k∈ℕ * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of R n,k , we also show that the space of indecomposable elements of 𝔽 q [x 1 ,...,x n ] in degree q n-1 -n has dimension equal to that of a complex cuspidal representation of GL n (𝔽 q ), that is (q-1)(q 2 -1)⋯(q n-1 -1).

Butson full propelinear codes

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the k{mathrm{th}} roots of unity, we can construct a larger Butson matrix over the ell mathrm{th} roots of unity for any ell dividing k, provided that any prime p dividing k also divides ell . We prove that a {mathbb {Z}}_{p^s}-additive code with p a prime number is isomorphic as a group to a BH-code over {mathbb {Z}}_{p^s} and the image of this BH-code under the Gray map is a BH-code over {mathbb {Z}}_p (binary Hadamard code for p=2). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.

Polynomial functions over dual numbers of several variables

Let [Formula: see text] be a positive integer. For a commutative ring [Formula: see text], the ring of dual numbers of [Formula: see text] variables over [Formula: see text] is the quotient ring [Formula: see text], where [Formula: see text] is the ideal generated by the set [Formula: see text]. This ring can be viewed as [Formula: see text] with [Formula: see text], where [Formula: see text] for [Formula: see text]. We investigate the polynomial functions of [Formula: see text] whenever [Formula: see text] is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on [Formula: see text] depending on the order of the pointwise stabilizer of the subring of constants [Formula: see text] in the group of polynomial permutations of [Formula: see text]. Further, we show that the stabilizer group of [Formula: see text] is independent of the number of variables [Formula: see text]. Moreover, we prove that a function [Formula: see text] on [Formula: see text] is a polynomial function if and only if a system of linear equations on [Formula: see text] that depends on [Formula: see text] has a solution.

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 Nonlinear Rotordynamics Model for Automotive Turbochargers Coupled to a Physical Model for a (SEMI) Floating Ring Bearing System

Abstract Automotive turbochargers (TCs) use an engine oil lubricated bearing system to produce acceptable performance (as per the engine volumetric efficiency) and proven reliability. However, the bearings also cause TC rotordynamic responses that are rich in subsynchronous whirl motions through reaching stable limit cycles. The paper describes the lubrication model for a finite length semi-floating ring bearing (SFRB) system and its coupling to the rotor and ring structure models for prediction of both linear and nonlinear system responses and their characterization in terms of motion amplitudes and whirl frequency content. The SFRB model includes a thermal energy transport network for the inner and outer films in both radial bearings and the thrust bearings located on the end sides of the ring. The large temperature difference between the hot shaft and a cold housing induces a three-dimensional thermal gradient in the fluid films and the floating ring, further exacerbated by the heat generated from drag power losses in the inner films adjacent to the rotor. The temperature gradients affect the lubricant viscosity and the bearing system operating clearances. The integration of the rotor and bearing system (RBS) equations of motion accounts for the SFRB nonlinear forces and starts from static equilibrium position, if existing. The model predictions of nonlinear behavior are accurate when benchmarked to a set of measurements procured in a gas stand test rig. The analysis also investigates the influence of the bearing inner clearance and rotor mass imbalance distribution on the onset, persistence and severity of SSV.

QSCs from the Euclidean sums of cyclic codes over finite rings

QSCs from the Euclidean sums of cyclic codes over finite rings

Extensions of Schönemann's theorem in Galois rings

Let p be a prime and Fp be a finite field of p elements. For a1,...,ak,b∈Fp, let NFp(a1,...,an,b) denote the number of the solutions (x1,...,xn)∈Fpn to the following linear equationsa1x1+⋅⋅⋅+anxn=b with the constraint xi≠xj for all i≠j. In 1839, Schönemann obtained the exact formula of NFp(a1,...,an,0) provided that ∑i=1nai=0 and ∑i∈Iai≠0 for all I⫋{1,...,n}. Schönemann's theorem was generalized by Bibak, Kapron and Srinivasan from Fp to Z/mZ for any positive integer m. Recently, Li and Yu generalized Schönemann's theorem to the finite field Fq with general prime power q. In this article, we give a further extension of Schönemann's theorem in the Galois rings GR(p2,r).

On the condition that powers preserve noncentralness

Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products.

Non-Trivial Subring Perfect Codes in Unit Graph of Boolean Rings

The aim of this paper is to investigate the non-trivial subring perfect codes in a unit graph associated with the Boolean rings. We prove a subring perfect code of size , where , in the unit graphs associated with the finite Boolean rings . Moreover, we give a necessary and sufficient condition for a subring of an infinite Boolean ring to be a perfect code of size infinity in the unit graph.

On networks over finite rings

The (control) networks over finite rings are proposed and their properties are investigated. Based on semi-tensor product (STP) of matrices, a set of algebraic equations are provided to verify whether a finite set with two binary operators is a ring. As an application, all rings with 4 elements are obtained. It is then shown that the STP-based technique developed for logical (control) networks are applicable to (control) networks over finite rings. A (control) sub-network over a proper ideal of the bearing ring is constructed. Certain properties are revealed, showing that a network over the ideal behaves like a subsystem over an invariant subspace. Product rings are then introduced, which provides a tool for both constructing product networks and decomposing complex networks. Finally, the representation of networks over finite rings is considered, which investigates how many finite networks can be expressed as networks over finite rings, showing that the technique developed in this paper is widely applicable.

Characterizations and Properties of Monic Principal Skew Codes over Rings

Let A be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, δ a σ -derivation of A , and consider the skew-polynomial ring A X ; σ , δ . When A is a finite field, a Galois ring, or a general ring, some fairly recent literature used A X ; σ , δ to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal f , σ , δ -codes, where f ∈ A X ; σ , δ is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code C . When A is a finite commutative ring and σ is a ring automorphism of A , we also give recursive formulas for the entries of a parity-check matrix of C . Also, in this case, with δ = 0 , we present a characterization of monic principal σ -codes whose dual codes are also monic principal σ -codes, and we deduce a characterization of self-dual monic principal σ -codes. Some corollaries concerning monic principal σ -constacyclic codes are also given, and a good number of highlighting examples is provided.