Search

In this article, based on the defining set $$D=\{x\in \mathbb {F}_{p^m}^*:Tr(x)=0\}$$ , we explore the Lee-weight distribution of linear codes $${\mathcal {C}}_D=\{(tr(ax^2))_{x\in D}:a\in \mathbb {F}_{p^m}+u\mathbb {F}_{p^m}\}$$ over the finite ring $$\mathbb {F}_p+u\mathbb {F}_p$$ with p being an odd prime and $$u^2=0$$ . By employing the exponential and Gauss sums, we calculate the Lee weight of all possible codewords as well as their frequencies. Two classes of eight-weight linear codes are obtained, where one of them is new. We also show that for some small values m, the code $${\mathcal {C}}_D$$ has two weights ( $$m=2$$ ) and seven weights ( $$m=3,4$$ and $$p=3$$ ).

In this paper, first we consider the linear complexity of quaternary sequences over the finite ring of order four and the finite field of order four. These sequences are constructed from new generalized cyclotomic classes modulo pn. Second, we study the linear complexity of new generalized cyclotomic binary sequences of period 2pn recently proposed by Ouyang et al. (Des. Codes Cryptogr. 87(5), 1–12 2019). We generalized results presented in this work and discuss the author’s conjecture. In conclusion, we again derive the linear complexity of quaternary sequences with period 2pn over the finite ring of order four and the finite field of order four. We use generalized cyclotomic classes modulo 2pn to define these sequences.

Secret sharing is an important concept in cryptography, however it is a difficult problem to determine the access structure of the secret sharing scheme based on a linear code. In this work, we construct two-weight linear codes over finite field by using linear codes over finite ring. We first study MacDonald codes over the finite ring $\mathbb{F}_2$+v$\mathbb{F}_2$+v2$\mathbb{F}_2$ with v3=v. Then we give torsion codes of MacDonald codes of type α and β, which are two-weight linear codes. Finally we give the access structures of secret sharing schemes based on the dual codes of the two-weight codes.

We find bounds for the number of idempotents in a ring of matrices over an arbitrary finite commutative ring. Using these results, we find a bound on the number of idempotents in an arbitrary finite ring, whereby we improve upon the currently known bounds for the number of idempotents.

We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.

Von Neumann regularity within a ring of upper triangular matrices is characterized in terms of how such a matrix and an upper triangular generalized inverse can be constructed. In the case of a fin...

Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.

In this paper, we establish a mass formula for Euclidean and Hermitian self-orthogonal codes over the finite ring Fq + uFq, where Fq is the finite field of order q and u2 = 0. We also establish a mass formula for Euclidean self-orthogonal codes over the finite ring Fq + uFq + u2Fq, with u3 = 0 and characteristic of Fq is odd. These mass formulas are used to give a classification of Euclidean and Hermitian self-orthogonal codes over F2 + uF2 and F3 + uF3 of small lengths.

The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite ring.

Nearly $60$ years ago, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which $2$-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all $2$-groups of exponent $4$ and exponent $2$ are realizable in characteristic $2$, and we prove that many $2$-groups with exponent $4$ and nilpotency class $3$ are realizable in characteristic $2$. On the other hand, we provide an example of a $2$-group with exponent $4$ and nilpotency class $4$ that is not realizable in characteristic $2$. Moreover, while some groups of exponent greater than $4$ are realizable as unit groups of rings, we prove that any $2$-group with a self-centralizing element of order $8$ or greater is never realizable in characteristic $2^m$, and consequently any indecomposable, nonabelian group with a self-centralizing element of order $8$ or greater cannot be the group of units of a finite ring., Comment: 27 pages. The main theorem in the previous version was incorrect; specifically, Proposition 3.9 was incorrect. Pages 9-13 of this version include new content dedicated to the proof of Theorem 1.2 (the updated main result), and Section 4 is dedicated to a counterexample to the main theorem in the previous version. Question 6.8 is also new

A characterization of idempotency is given for triangular matrices with entries from an arbitrary ring. This yields an elementary approach to constructing such matrices. It also reveals structure a...

In this paper, the exact value of covering radius of unit repetition codes and the bounds of covering radius of zero-divisor repetition codes have been determined by using Lee weight over the finite ring \(F_2+vF_2+v^{2}F_2\). Moreover the covering radius of different block repetition codes have been also studied.

Associative rings are considered. By a lattice isomorphism, or projection, of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of R onto the subring lattice L(Rφ) of Rφ. In this case Rφ is called the projective image of a ring R and R is called the projective preimage of a ring Rφ. Let R be a finite ring with identity and Rad R the Jacobson radical of R. A ring R is said to be local if the factor ring R/Rad R is a field. We study lattice isomorphisms of finite local rings. It is proved that the projective image of a finite local ring which is distinct from GF($$ {\mathrm{p}}^{{\mathrm{q}}^{\mathrm{n}}} $$) and has a nonprime residue field is a finite local ring. For the case where both R and Rφ are local rings, we examine interrelationships between the properties of the rings.

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A⊂B is a decomposed (m...

Let R be a finite ring. The zero-divisor graph of R, written as G(R), is a digraph with vertex set Z*(R), the set of all nonzero zero-divisor of R, and for x,y∈Z*(R) (needless distinct) there exist...

Let R be a commutative ring with a proper ideal I . A generalization of total graph is introduced and investigated. It is the (undirected) graph with all elements of R as vertices, that two distinct vertices x , y ∈ R are adjacent if and only if x + y ∈ S H ( I ) where S H ( I ) = { a ∈ R : r a ∈ I for some r ∈ H } and H is a multiplicatively closed subset of R . This version of total graph is denoted by T ( Γ H I ( R ) ) . We in addition characterize certain lower and upper bounds for the genus of the total graph, and compute genus T ( Γ H I ( R ) ) on finite ring R , with respect to some special ideal I .

In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.

In this work, we investigate DNA codes of odd length over the finite ring R = Z 4 +vZ 4 , v 2 = v. Firstly, we give a map Φ from R n to {A, T, G, C} 2n . Then cyclic codes of odd length satisfying the reverse and reverse-complement constraint are characterized over such ring. Moreover, when C is a reverse-complement cyclic code over R, we prove that Φ(C) is a cyclic DNA code by using the method different from all the previously known ones, which helps us finding a new DNA code with 256 codewords. Finally, we consider the GC content of code C over finite ring Z 4 + vZ 4 by means of the Gray images of the minimal generating set of C.

WOS: 000531286600021 The DNA 4-bases are matched with the 256 elements of the finite ring R=F4+uF4+vF4+uvFR=F_{4}+uF_{4}+vF_{4}+uvF_{4}$$\end{document}, where u2=u,v2=v,uv=vu. By defining a non trivial automorphism over R, the skew cyclic codes over the finite ring R are introduced. The reversible DNA codes are obtained, by using the skew cyclic codes whose generator polynomials satisfy some conditions. Thanks to this, the reversibility problem for DNA 4-bases can be solved. Moreover, the Gray images of the skew cyclic codes over the finite ring are determined.

Let p be an odd prime and r a positive integer. Let GR(p3, r) be the Galois ring of characteristic p3 and cardinality p3r. In this paper, we investigate the self-dual codes over GR(p3, r) and give a method to construct self-dual codes over this ring. We establish a mass formula for self-dual codes over GR(p3, r) and classify self-dual codes over GR(p3, 2) of length 4 for p = 3, 5.

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born–von Karman boundary conditions. A second formula, alternative to Resta’s, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.

Let F3d is the finite field of order 3d with d be a positive integer, we consider A4:=F3d[e]=F3d[X]/(X4) is a finite quotient ring, where e4=0. In this paper, we will show an example of encryption and decryption. Firstly, we will present the elliptic curve over this ring. In addition, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. Precisely, we give a numerical example of cryptography (encryption and decryption) by using two methods with a secret key.

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \le 5$. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree.

In this paper, the probability that two elements of a finite ring have product zero is considered. The bounds of this probability are found for an arbitrary finite commutative ring with identity 1. An explicit formula for this probability in the case of, the ring of integers modulo, is obtained.

For a ring R, let J(R) and C(R) be the Jacobson radical and the center of R, respectively. It is proved that a left uniserial, centrally essential finite ring is commutative. In the proof of this result, we use the following more general assertion: if F is a field, then any left uniserial, left Artinian, centrally essential ring R with factor field $$R/J(R)\cong F$$ is commutative if and only if the field F does not have non-zero derivations. It is proved that there are non-commutative uniserial Artinian centrally essential rings and right uniserial, right Noetherian PI rings which are not centrally essential. It is also proved that a left Artinian, left uniserial ring R with J(R) of nilpotence index n is centrally essential provided $$\displaystyle {J(R)^{\left[ n/2\right] }\subseteq C(R)}$$.

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly ...

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that |B|≤N for each ring B such that A⊂B is an inert (minimal ring) extension. It foll...

Pairwise weakly linearly independent (PWLI) sets were introduced in order to compute sharp cyclic coverings of a finite module over a finite ring. In this work, new relationships between PWLI sets and short coverings of the linear space F q 3 are explored. As a consequence, we investigate the minimum cardinality of a short covering of F q 3 when q is odd, in order to improve a previous upper bound.

We prove a formula for the number of representations of an element in a finite basic ring as a sum of k exceptional units and find bounds for this number in an arbitrary finite ring with identity.

Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\{O,E,O\oplus E\}$ , where $O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

Maximum distance profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of $p$ -encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over ${\mathbb Z}_{p^{r}}$ is presented.

The article is devoted to the problem of creation of high-speed neural networks (NN) for calculation of interval-index characteristics of a minimally redundant modular code. The functional base of the proposed solution is an advanced class of neural networks of a final ring. These neural networks perform position-modular code transformations of scalable numbers using a modified reduction technology. A developed neural network has a uniform parallel structure, easy to implement and requires the time expenditures of the order (3[log2b]+ [log2k]+6tsum close to the lower theoretical estimate. Here b and k is the average bit capacity and the number of modules respectively; t sum is the duration of the two-place operation of adding integers. The refusal from a normalization of the numbers of the modular code leads to a reduction of the required set of NN of the finite ring on the (k – 1) component. At the same time, the abnormal configuration of minimally redundant modular coding requires an average k-fold increase in the interval index module (relative to the rest of the bases of the modular number system). It leads to an adequate increase in hardware expenses on this module. Besides, the transition from normalized to unregulated coding reduces the level of homogeneity of the structure of the NN for calculating intervalindex characteristics. The possibility of reducing the structural complexity of the proposed NN by using abnormal intervalindex characteristics is investigated.

The rate of a network code is the ratio of the block size of the network's messages to that of its edge codewords. We compare the linear capacities and achievable rate regions of networks using finite field alphabets to the more general cases of arbitrary ring and module alphabets. For non-commutative rings, two-sided linearity is allowed. Specifically, we prove the following for directed acyclic networks: (i) The linear rate region and the linear capacity of any network over a finite field depend only on the characteristic of the field. Furthermore, any two fields with different characteristics yield different linear capacities for at least one network. (ii) Whenever the characteristic of a given finite field divides the size of a given finite ring, each network's linear rate region over the ring is contained in its linear rate region over the field. Thus, any network's linear capacity over a field is at least its linear capacity over any other ring of the same size. An analogous result also holds for linear network codes over module alphabets. (iii) Whenever the characteristic of a given finite field does not divide the size of a given finite ring, there is some network whose linear capacity over the ring is strictly greater than its linear capacity over the field. Thus, for any finite field, there always exist rings over which some networks have higher linear capacities than over the field., Comment: Submitted to IEEE Transactions on Information Theory June 4th, 2017 Revised January 29th, 2018

We present a new and simpler proof of a combinatorial nature for the commutative case of Wood’s theorem which states that a finite ring has a generating character if and only if it is a Frobenius ring.

Data encryption has become a vital mechanism for data protection. One of the main challenges and an important target for optimization is the encryption/decryption speed. In this paper, we propose techniques for speeding up the software performance of several important cryptographic primitives based on the Residue Number System (RNS) and Finite Ring Neural Network (FRNN). RNS&FRNN reduces the computational complexity of operations with arbitrary-length integers such as addition, subtraction, multiplication, division by constant, Euclid division, and sign detection. To validate practical significance, we compare LLVM library implementations with state-of-the-art, high-performance, portable C++ NTL library implementations. The experimental analysis shows the superiority of the proposed optimization approach compared to the available approaches. For the NIST FIPS 186-5 digital signature algorithm, the proposed solution is 85% faster, even though the sign detection has low efficiency.

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn=FqX/Xn, where n≥1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves.

Let [Formula: see text] be a unitary ring of finite cardinality [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text]. We show that if the group of units of [Formula: see text] has at most one subgroup of order [Formula: see text], then [Formula: see text] where [Formula: see text] is a finite ring of order [Formula: see text] and [Formula: see text] is a ring of cardinality [Formula: see text] which is one of the six explicitly described types.

We find a building-up type construction method for self-orthogonal codes over Z 4 arising from the chain ring Z 4 [ u ] / 〈 u 2 + 1 〉 . Our construction produces self-orthogonal codes over Z 4 with increased lengths and free ranks from given self-orthogonal codes over Z 4 with smaller lengths and free ranks; in the most of the cases their minimum weights are also increased. Furthermore, any self-orthogonal code over Z 4 with generator matrix subject to certain conditions can be obtained from our construction. Employing our construction method, we obtain at least 125 new self-orthogonal codes over Z 4 up to equivalence; among them, there are 35 self-orthogonal codes which are distance-optimal. Furthermore, we have eight self-orthogonal codes, which are distance-optimal among all linear codes over Z 4 with the same type. As a method, we use additive codes over the finite ring Z 4 [ u ] / 〈 u 2 + 1 〉 with generator matrices G satisfying G G T = O , and we use a new Gray map from Z 4 [ u ] / 〈 u 2 + 1 〉 to Z 4 3 as well.

In this paper, we generalize the notion of self-orthogonal codes to \begin{document}$ \sigma $\end{document}-self-orthogonal codes over an arbitrary finite ring. Then, we study the \begin{document}$ \sigma $\end{document}-self-orthogonality of constacyclic codes of length \begin{document}$ p^s $\end{document} over the finite commutative chain ring \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}, where \begin{document}$ p $\end{document} is a prime, \begin{document}$ u^2 = 0 $\end{document} and \begin{document}$ \sigma $\end{document} is an arbitrary ring automorphism of \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. We characterize the structure of \begin{document}$ \sigma $\end{document}-dual code of a \begin{document}$ \lambda $\end{document}-constacyclic code of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. Further, the necessary and sufficient conditions for a \begin{document}$ \lambda $\end{document}-constacyclic code to be \begin{document}$ \sigma $\end{document}-self-orthogonal are provided. In particular, we determine all \begin{document}$ \sigma $\end{document}-self-dual constacyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. In the end of this paper, when \begin{document}$ p $\end{document} is an odd prime, we extend the results to constacyclic codes of length \begin{document}$ 2 p^s $\end{document}.

The research on color Visual Cryptographic Scheme (VCS) is much more difficult than that of the black and white VCS. This is essentially because of the fact that in color VCS, the rule for superimposition of two colors is not that simple as in black and white VCS. It was a long standing open issue whether linear algebraic technique in constructing Black and White visual cryptographic schemes could also be extended for color images. It was thought that such an extension was impossible. However, we resolve this issue by providing color VCS in same color model for the threshold access structures by extending linear algebraic techniques from the binary field \(\mathbb {Z}_2\) to finite ring \(\mathbb {Z}_c\) of integers modulo c. We first give a construction method based on linear algebra to share a color image for an (n, n)-threshold access structure. Then we give constructions for (2, n)-threshold access structures and in general (k, n)-threshold access structures. Existing methodology for constructing color VCS in same color model assumes the existence of black and white VCS, whereas our construction is a direct one. Moreover, we give closed form formulas for pixel expansion which is combinatorially a difficult task. Lastly, we give experimental results and propose a method to reduce pixel expansion.

A cover of a unital, associative (not necessarily commutative) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ ( R ) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ ( R ) σ ( R / I ) for every nonzero two-sided ideal I of R. In this paper, we show that if R has a finite covering number, then the calculation of σ ( R ) can be reduced to the case where R is a finite ring of characteristic p and the Jacobson radical J of R has nilpotency 2. Our main result is that if R has a finite covering number and R / J is commutative (even if R itself is not), then either σ ( R ) = σ ( R / J ) , or σ ( R ) = p d + 1 for some d ⩾ 1 . As a byproduct, we classify all commutative σ-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring.

The study on probability theory in finite rings has been an interest of various researchers. One of the probabilities that has caught their attention is the probability that two elements of a ring have product zero. In this study, the probability is determined for a finite ring R of matrices over integers modulo four. First, the annihilators of R are determined with the assistance of Groups, Algorithms and Programming (GAP) software and then the probability is calculated using the definition. Next, by using the results obtained, the zero divisor graph of the ring R is constructed. A zero divisor graph is defined as a graph in which the zero divisors of R are its vertices and two vertices are connected by an edge if their product is zero.

In this article, for any odd prime p, we study the cyclic codes over the finite ring \(R=\mathbb {F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1,vw-wv\rangle \) to obtain the quantum codes over \(\mathbb {F}_{p}\). We obtain the necessary and sufficient condition for cyclic codes which contain their duals and as an application, some new quantum codes are presented at the end of the article.

Let D be a subset of a finite commutative ring R with identity. Let $$f(x)\in R[x]$$ be a polynomial of degree d. For a nonnegative integer k, we study the number $$N_f(D,k,b)$$ of k-subsets S in D such that $$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$ In this paper, we establish several bounds for the difference between $$N_f(D,k, b)$$ and the expected main term $$\frac{1}{|R|}{|D|\atopwithdelims ()k}$$ , depending on the nature of the finite ring R and f. For $$R=\mathbb {Z}_n$$ , let $$p=p(n)$$ be the smallest prime divisor of n, $$|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c$$ and $$f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]$$ with $$(a_d, \ldots , a_1, n)=1$$ . Then $$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$ answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where $$\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}$$ and $$C_d=e^{1.85d}$$ . Furthermore, if n is a prime power, then $$\delta (n) =1/p$$ and one can take $$C_d=4.41$$ . Similar and stronger bounds are given for two more cases. The first one is when $$R=\mathbb {F}_q$$ , a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.

Quantum codes over finite rings have received a great deal of attention in recent years. Compared with quantum codes over finite fields, a notable advantage of quantum codes over finite rings is that they can adapt to quantum physical systems of arbitrary order. Moreover, operations are much easier to execute in finite rings than they are in fields. The modulo $m$ residue class ring $\mathbb {Z}_{m}$ is the most common finite ring. This paper investigates stabilizer codes over $\mathbb {Z}_{m}$ and presents the Gilbert–Varshamov (GV) bound. The GV bound shows that surprisingly good quantum codes exist over $\mathbb {Z}_{m}$ , which makes quantum coding feasible for arbitrary quantum physical system. We also provide an enhanced version of the GV bound for non-degenerate stabilizer codes over $\mathbb {Z}_{m}$ . The enhanced GV bound has an asymptotical form and ensures the existence of asymptotically good stabilizer codes over $\mathbb {Z}_{m}$ . Finally, these two bounds are well suited for computer searching.

Let R be a finite commutative ring with identity. In this paper, given an integer $$k\ge 2$$ , we obtain an exact formula for the number of ways to represent each element of R as the sum of k exceptional units. This generalizes a recent result of Quan-Hui Yang and Qing-Qing Zhao for the case where R is the ring $${\mathbb {Z}}_n$$ of residue classes modulo n.

In this paper, we establish a mass formula for self-dual codes over the finite chain ring $$\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$$ , where $$\mathbb {F}_q$$ is the finite field of order q and $$u^3=0$$ . We also give a classification of self-dual codes over $$\mathbb {F}_q+u\mathbb {F}_q+u^2\mathbb {F}_q$$ , for $$q=2,3,4,5,7,8,9$$ , with lengths 2 and 4.