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Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: 'Classification of singularities of curves can be interpreted in dual terms as a description of "co-artin" subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)'. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let K be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group Aut K (A) is explicitly described. The automorphism group of the algebra A is finite if and only if the algebra A is not isomorphic to a monomial algebra, and in this case | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian 2-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Temperley–Lieb algebras TLn(δ) at δ = 1. Since δ = 1, we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a topological space, we get an algebra action on each homology group. The main theorem of this paper explicitly deduces the representation structure of the homology groups in terms of a natural filtration associated with our TLn-space. As a corollary of this result, we are able to study stability phenomena. There is a natural way to define representation stability in the context of TLn(1), and the presence of filtrations enables us to define a notion of topological stability. We are able to deduce that a filtration-stable sequence of TLn-spaces results in representation-stable sequence of homology groups. This can be thought of as the analogue of the statement that the homology of configuration spaces forms a finitely generated FI-module. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x,y are adjacent if x,y ∈〈z〉 for some z ∈ G. An L(2, 1)-labeling of graph Γ is an integer labeling of V (Γ) such that adjacent vertices have labels that differ by at least 2 and vertices distance 2 apart have labels that differ by at least 1. The λ-number of Γ, denoted by λ(Γ), is the minimum range over all L(2, 1)-labelings. In this paper, we study the lambda number of the enhanced power graph 풫E(G) of the group G. This paper extends the corresponding results, obtained in [X. Ma, M. Feng and K. Wang, Lambda number of the power graph of a finite group, J. Algebraic Combin. 53(3) (2021) 743–754], of the lambda number of power graphs to enhanced power graphs. Moreover, for a nontrivial simple group G of order n, we prove that λ(풫E(G)) = n if and only if G is not a cyclic group of order n ≥ 3. Finally, we determine the lambda number of the enhanced power graphs of nilpotent groups. [ABSTRACT FROM AUTHOR]

In the this paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too. This includes, for instance, algebraic algebras, group rings, and matrix rings over commutative rings. Moreover, we also obtain some independent results for the new class of rings. For example, the triangular matrix rings retain that property. [ABSTRACT FROM AUTHOR]

Let G = 핊d, d = 0, 1 or 3, act freely on a finitistic connected space X. This paper gives the cohomology classification of X if the mod 2 or rational cohomology of the orbit space X/G is isomorphic to the product of a projective space and sphere 픽ℙn קm, where 픽 = ℝ, ℂ or ℍ, respectively. For a free involution on X, a lower bound of covering dimension of the coincidence set of a continuous map f : X → ℝk is also determined. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a weighted derivation on algebras over an operad 풫, and prove that for the free 풫-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential (q-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential (q-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts. [ABSTRACT FROM AUTHOR]

Let G be a group with identity and R be a G-graded commutative ring with nonzero identity. A proper graded ideal I of R is called a graded 1-absorbing prime ideal (respectively, graded 1-absorbing primary ideal) if whenever nonunit homogeneous elements a,b,c ∈ R with abc ∈ I, then ab ∈ I or c ∈ I (respectively, ab ∈ I or c ∈ Gr(I), where Gr(I) is the graded radical of I). The purpose of this paper is to study the transfer of some graded 1-absorbing-like properties to the graded amalgamated algebra along an ideal (denoted by A⋈fJ). Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties. [ABSTRACT FROM AUTHOR]

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be an ICΦ-subgroup of G if H ∩ [H,G] ≤ Φ(H). In this paper, based on research work of Kaspczyk, we analyze p-hypercyclically embedded structure of a finite group G under the assumption that some given p-subgroups of G are ICΦ-subgroups of G. Also, all ICΦ-subgroups of symmetric group Sn are determined. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the well-known semigroup property of being E-solid is actually equivalent to abelianness modulo the Jacobson radical in the case of (possibly non-unital) non-regular rings. We also give various other characterizations of such rings. [ABSTRACT FROM AUTHOR]

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings R are always "J-abelian" in the sense that each idempotent of R maps onto a central idempotent in R /rad(R). On the other hand, J-abelian rings turn out to be precisely the "strongly perspective rings" as well as the "strongly IC rings", and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking n -th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings. [ABSTRACT FROM AUTHOR]

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length n p s over p m , where p is a prime number and p does not divide n , the problem has been solved completely for all n ≤ 6 and partially for n = 7 , 8. In this paper, we solve the problem for n = 9 and all primes p different from 3 and 1 9. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length 9 p s over p m . As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes. [ABSTRACT FROM AUTHOR]

We provide three generalizations of all best known generating sets for S n and A n : (1) m -cycles with consecutive entries; (2) m -cycles with fixed first ℓ entries; (3) two cycles, either (1 , ... , m) , (1 , ... , n) or (1 , ... , m) , (2 , ... , n). Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767]. [ABSTRACT FROM AUTHOR]

Hypercyclic rings are those rings over which cyclic modules have cyclic injective hulls. Such rings were introduced by Caldwell in 1966 and studied by him and Osofsky in 1968. In this paper, we revisit Osofsky's work on hypercyclic rings, highlighting some of her results and questions on the subject. [ABSTRACT FROM AUTHOR]

In this paper, we report on a computation demonstrating that if A and B are nonempty subsets of a torsion-free group such that A B has no unique product, then | A | + | B | ≥ 1 6. Moreover, this bound is sharp, as there are examples where | A | = | B | = 8 , and in fact A = B for at least two such examples. More generally, when | A | is small, we find lower bounds on | B |. One consequence of this work is that any counterexample to Kaplansky's zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate polyform modules and introduce the notion of relatively polyform modules over a ring R. Some new characterizations and properties of polyform modules are obtained. For a retractable polyform module M R , it is proved that the maximal right ring of quotients of End R (M) is End R (E ̃ (M)) , where E ̃ (M) is the rational hull of M. Moreover, we characterize nonsingular modules by using the relatively polyform property. In addition, it is proved that the class of all polyform right R -modules is closed under submodules and rational extensions. [ABSTRACT FROM AUTHOR]

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual. [ABSTRACT FROM AUTHOR]

Let be an ∗-algebra and k ∈ ℤ + . For A , B ∈ , the product defined by ∗ [ A , δ A (B) λ ∗ ] k = ∗ [ A , ∗ [ A , δ A (B) λ ∗ ] k − 1 ] 1 , where ∗ [ A , δ A (B) λ ∗ ] 0 = δ A (B) λ ∗ and ∗ [ A , δ A (B) λ ∗ ] 1 = A δ A (B) λ ∗ − δ A (B) λ ∗ A ∗ , is called λ - k -skew Lie product of A and B , where δ A (B) λ ∗ = A B − λ B A ∗ . This paper discusses some useful characteristics of λ - k -skew Lie products on prime ∗-algebras. Moreover, we investigate the effects of an additive map ϕ on with a λ - k -skew Lie product. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we shall study three types of Galois extensions B which are Azumaya algebras over its center C. (1) 1 , the class of Azumaya–Galois extensions; (2) 2 , the class of Galois extensions B of B G with Galois group G such that B G is a separable C G -algebra; and (3) 3 , the class of Galois extensions which are Azumaya algebras over its center C. Clearly, 1 ⊆ 2 ⊆ 3 . Examples are given to show the proper inclusion relationship 1 ⊂ 2 ⊂ 3 , and equivalent conditions are given under which a B ∈ 3 is in 1 and a B ∈ 2 is in 1 . [ABSTRACT FROM AUTHOR]

The well-known notion of an extending module is closely linked to that of a Baer module. A right R -module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R -module M is called Baer if for all N ≤ M , l S (N) ≤ ⊕ S S where S = End R (M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is -nonsingular extending if and only if M is -cononsingular Baer". M R is called -nonsingular if ∀ φ ∈ S such that Ker φ ≤ e M , φ = 0. Moreover, M R is called -cononsingular if for any N ≤ M with φ N ≠ 0 for all 0 ≠ φ ∈ S , implies N ≤ e M. In view of this result, every Baer module which happens to be -cononsingular will automatically become an extending module. In this paper, our main focus is the study of -cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of -cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of -cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring of characteristic different from 2 , with right Martindale quotient ring Q r and extended centroid C. Let m , n ≥ 1 be fixed integers and F a nonzero generalized skew derivation of R. In this paper, we investigate the set S = { F (x m) x n + x n F (x m) : x ∈ R } and prove that its left annihilator in R is identically zero. Using the above result, we also study the identity F (x) x n + x n F (x) = 0 for semiprime rings. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

A module M is said to be extending (Goldie extending) if for each submodule X ≤ M , there exists a direct summand D of M such that X is essential in D (if for every submodule X of M there exists a direct summand D of M such that X ∩ D is essential in both X and D). In this paper, a necessary and sufficient condition is obtained for a trivial Morita context ring to be right Goldie-extending. [ABSTRACT FROM AUTHOR]

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring D [ x , σ , δ ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D [ x ]. In particular, we show that if a ∈ D and f ∈ D [ x , σ , δ ] satisfy f (a) = a , then f ∘ n (a) = a for every formal power of f. More generally, we give a sufficient condition for a point a to be r -periodic with respect to a polynomial f. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy. [ABSTRACT FROM AUTHOR]

In this paper, we classify the gradings up to equivalence on the filiform Lie superalgebras Lm,n over an algebraically closed field of characteristic 0. Moreover, by the relation between a grading and a standard grading, we describe all pairwise inequivalent gradings and their universal groups of Lm,n. [ABSTRACT FROM AUTHOR]

Let R be a commutative integral domain with unity that contains ℚ. In this paper we describe the R-automorphism group of R-algebras of the form R[x,y,z]/(c(x)z − q(x,y)), where c(x) ∈ R[x] is quasi-monic and q(x,y) ∈ R[x,y] is quasi-monic of degree at least two with respect to y. [ABSTRACT FROM AUTHOR]

Let 픛 be a class of finite groups. A group G is called a minimal non-픛-group or simply an 픛-critical group, if G is not in 픛 but all proper subgroups of G are in 픛. An 픑-critical group, for the class 픛 = 픑 all finite nilpotent groups, is called a Schmidt group. We say that a formation 픉 has the Shemetkov property if every 픉-critical group is either a Schmidt group or a cyclic group of prime order. In this paper, properties of the lattice of all soluble subgroup-closed local formations with Shemetkov property are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a condition on a p-subgroup of a finite group that is more general than the concept of ICΦ-subgroups, and obtain a sufficient and necessary condition for p-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results. [ABSTRACT FROM AUTHOR]

For any group G, let cent(G) denote the set of all centralizers of G. K. Khoramshahi and M. Zarrin [Groups with the same number of centralizers, J. Algebra Appl. 20 (2021) 2150012], posed the following conjecture: Let G and S be finite groups. Is it true that if |Cent(G)| = |Cent(S)| and |G′| = |S′|, then G is isoclonic to S? In this paper, among other things, we give a negative answer to this conjecture. [ABSTRACT FROM AUTHOR]

The two-dimensional systems given by complex numbers ℂ (i2 = −1), dual numbers 픸 (휀2 = 0) and hyperbolic numbers 픻 (j2 = 1) are, up to algebra isomorphism, the three possible associative algebra structures on ℝ2. The goal of this work is to investigate canonical numbers systems for the rings of integers in the two-dimensional systems yielding digital representation systems in the plane. Kátai and Szabó [Canonical number systems for complex integers, Acta Sci. Math. 37 (1975) 255–260] proved that all complex numbers can be written in radix expansion with the natural numbers 0, 1,…,n2 as digits. In this paper, we will characterize all canonical number systems for the rings of integers in dual and hyperbolic numbers. Finally, using the associate matrices of the two-dimensional bases, we prove that all points in the plane ℝ2 can be written in digital representation systems with a large class of bases, including binary, octal, decimal and hexadecimal ones. In particular, we prove that a digital representation system in the plane is finite or periodic if and only if it represents a point with rational coordinates. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary ring equipped with an automorphism α and an α-derivation δ. In this paper, we study on the relationship between the piecewise prime property of a ring R and these of the skew inverse Laurent series ring R((x−1; α,δ)) and related rings. Furthermore, for a piecewise prime ring, we determine a large class of rings which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many available results. Examples for illustration and delimitation of the theory are provided. [ABSTRACT FROM AUTHOR]

An integral domain R is perinormal if every local going-down overring is a localization of R and globally perinormal if every going-down overring is a localization of R. In this paper, I introduce notions of graded perinormal and graded globally perinormal domains and show that many results obtained for perinormal and globally perinormal domains have graded analogs. I also give some results for descent of properties between a graded domain and its 0th graded component. [ABSTRACT FROM AUTHOR]

Multiplicative differential (and the corresponding c-differential uniformity) was introduced by Ellingsen et al. in [C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity, IEEE Trans. Inf. Theory 66(9) (2020) 5781–5789], which has attracted lots of attention. Functions with low c-differential uniformity over finite fields, especially the PcN and APcN functions, have been widely investigated due to their applications in cryptography. In this paper, we first compute the c-differential uniformity of two classes of permutation polynomials. For one of these, we explicitly determine the c-DDT entries. For the second type of function, we give bounds for its c-differential uniformity. Besides, several classes of PcN or APcN functions are presented by employing some known functions and the (generalized) AGW criterion. [ABSTRACT FROM AUTHOR]

This paper describes the primary decomposition of the edge ideal of the join of weighted oriented graphs in terms of the information from each of those graphs. Using this decomposition and induction, we derive formulas for the depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices. Furthermore, we provide upper bounds for the regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least one oriented edge for each graph, we provide exact formulas for their depth and regularity, as well as upper bounds for the regularity of ordinary powers of such an edge ideal. Some examples demonstrate that these upper bounds are sometimes attainable and sometimes strict. [ABSTRACT FROM AUTHOR]

Let G be a finite nilpotent group. We denote by γ(G) the number of conjugacy classes of noncyclic subgroups of G, Con(G) the number of conjugacy classes of subgroups of G. In this paper, we give the formula for γ(G), and prove that Con(G) − γ(G) ≥ 4 when the order of G is prime power. Moreover, we get the minimum of γ(G) according to t, where t is the number of distinct noncyclic Sylow subgroups of G. [ABSTRACT FROM AUTHOR]

In this paper, we study Betti splittings of cover ideals of bipartite graphs. We prove that if J ⊂ 한[x1,…,xn] is the cover ideal of a bipartite graph then the xi-partition of J is a Betti splitting for any i. We also prove that multigraded Betti numbers of any squarefree monomial ideal can appear in a certain part of multigraded Betti numbers of the cover ideal of a bipartite graph. [ABSTRACT FROM AUTHOR]

As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation that generalizes Novikov algebras. Such objects are called noncommutative Novikov algebras. We construct a monomial basis for a free noncommutative Novikov algebra. [ABSTRACT FROM AUTHOR]

Let R be a polynomial ring in n indeterminates with coefficients in a field K of characteristic p > 0, and 풟 be the ring of differential operators over R. We consider 픭 as a nonmaximal homogeneous prime ideal, and ∗E(R/픭) as a graded injective hull of R/픭. In this paper, we prove the generalized Eulerian property for the graded injective hull R/픭 as a 풟-module in positive characteristic. Furthermore, we show that the minimal injective resolution of F-finite module ends in an object which is the direct sum of indecomposable in graded case using some finiteness properties in prime characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code. [ABSTRACT FROM AUTHOR]

There is a non-unital ring I of order 4 defined by generators and relations as I = 〈 a , b | 2 a = 2 b = 0 , a 2 = b , a b = 0 〉. In this paper, we present special constructions of linear codes over I from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in I are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given. [ABSTRACT FROM AUTHOR]

The use of codes defined by sparse characteristic matrices, like QC-LDPC and QC-MDPC codes, has become an established solution to design secure and efficient code-based public-key encryption schemes, as also witnessed by the ongoing NIST post-quantum cryptography standardization process. However, similar approaches have been less fortunate in the context of code-based digital signatures, since no secure and efficient signature scheme based on these codes is available to date. The main limitation of previous attempts in this line of research has been the use of sparse signatures, which produces some leakage of information about the private key. In this paper, we propose a new code-based digital signature scheme that overcomes such a problem by publishing signatures that are abnormally dense, rather than sparse. This eliminates the possibility of deducing information from the sparsity of signatures, and follows a recent trend in code-based cryptography exploiting the hardness of the decoding problem for large-weight vectors, instead of its classical version based on small-weight vectors. We focus on one-time use and provide a first parameterization of the new scheme, the main advantages of which are very fast signature generation and verification, and public keys that are not too large for a code-based signature scheme. [ABSTRACT FROM AUTHOR]

In [D. R. L. Brown, Plactic key agreement (insecure?), J. Math. Cryptol.17(1) (2023) 20220010], a novel cryptographic key exchange technique was proposed using the plactic monoid, based on the apparent difficulty of solving division problems in that monoid. Specifically, given elements c , b in the plactic monoid, the problem is to find q for which q b = c , given that such a q exists. In this paper, we introduce a metric on the plactic monoid and use it to give a probabilistic algorithm for solving that problem which is fast for parameter values in the range of interest. [ABSTRACT FROM AUTHOR]

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions. [ABSTRACT FROM AUTHOR]

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤ p r , ℤ p r ((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over ((d)) , where is a field, when dealing with ℤ p r ((d)) there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called p -encoder. In this paper we solve this conjecture for a class of (2 , 1) convolutional codes over ℤ p 2 and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic p -encoder. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) 2 -linear code over 2 r . Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes invariant under a cyclic endomorphism T , called T -cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M , that we call M -cyclic codes, and when M is the companion matrix C f of a given nonzero polynomial f (x) we call them f -cyclic codes. The similarity relation between matrices helps us find connections between M -cyclic codes and f -cyclic codes, where f (x) is the minimal polynomial of M. The class of M -cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M -cyclic codes and ideals of the polynomial ring R f : = q [ x ] / 〈 f (x) 〉 , where f (x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b -dual of an M -cyclic code is an M ∗ -cyclic code, where M ∗ is the adjoint matrix of M with respect to b , and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes. [ABSTRACT FROM AUTHOR]

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes ⊆ ∏ i = 1 N q i over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity. [ABSTRACT FROM AUTHOR]