Starting from the representation of a function f(x, y) as a formal power series with Taylor coefficients fm,n, a formal series is set up for the implicit function y = y(x) so that f(x, y) = 0 and the coefficients of the series for y depend exclusively on the fm,n. The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.
Algebraic number theory has recently attracted significant interest due to its role in algebraic lattice theory and in the design of codes for applications in coding theory. Algebraic lattices have been useful in information theory, where the problem of constructing lattices over number fields with full diversity and maximal minimum product distance has been investigated, since these parameters are directly related to error probabilities over Rayleigh fading channels. In this paper, we present a family of full diversity rotated unimodular lattices constructed via totally real subfields of the cyclotomic fields ℚ(ζp), with p an odd prime. A closed-form expression for the minimum product distance is derived.
It is known that threshold graphs have edge rings with 2-linear resolutions. This was proved by Engström and Stamps, [4]. They used the fact that an edge ring of a graph G has a 2-linear resolution if and only if the complement graph is chordal. They also described a method to determine the Betti numbers. Our goal is to determine when edge rings of threshold graphs are Cohen-Macaulay. In order to do so, it is more convenient to use an alternative way to study edge rings of graphs, that is to interpret them as Stanley-Reisner rings. We also determine when the neighborhood complex of a threshold graph has a Cohen-Macaulay Stanley-Reisner ring.
This article studies Θt-cyclic and (Θt, λ)-constacyclic codes over the finite commutative non-chain Frobenius ring R = Fq[u, v, w] / 〈 u2 − u, v2 − v, w2 − 1, uv, uw − wu, wv − vw 〉. Gray maps, structural decompositions, and generator descriptions are developed for both odd- and even-characteristic cases. The paper further determines principal generators in the associated skew polynomial rings, dual codes, idempotent generators, and conditions for self-duality. It also presents explicit examples over specific finite fields and extends the framework to DNA codes in the even-characteristic setting through reversibility and complement constraints. Spanning sets, cardinality formulas, and optimal DNA-code constructions meeting the Griesmer bound are also obtained.
In this paper, we study polycyclic codes over the ring Rl = Fq[w] / 〈wl − 1〉 with q = pk, where k is a positive integer and p is an odd prime. We explore LCD annihilator, self-dual, self-orthogonal codes over Rl and polycyclic codes over Rl. Moreover, we provide a structure of entanglement-assisted quantum error-correcting codes based on the developed polycyclic codes through the inclusion of dual contained conditions. Subsequently, some LCD hull code-derived entanglement-assisted quantum error-correcting code examples are presented.
This paper investigates oriented ℱ-designs on complete uniform hypergraphs of rank 3, focusing in particular on the spectrum of existence and on the construction of some cyclically and transitively oriented P(3)(2, 4)-designs and P(3)(1, 5)-designs, namely BCP(3)(2, 4)-designs and BCP(3)(1, 5)-designs for cyclically oriented ones, BTP(3)(2, 4)-designs and BTP(3)(1, 5)-designs for transitively oriented ones. In the appendix, we provide the Python code to obtain the explicit realization of the BCP(3)(2, 4)-designs on v vertices. Moreover, the structure of this algorithm, with suitable modifications, can be generalized to the other three structures as well.
Let Cn and Sn respectively denote a cycle and star with n edges. Let Kn denote a complete graph on n vertices. In this paper, it is shown that for any non-negative integers α and β and any positive integer n ≥ 6, there exists a decomposition of Kn into α copies of C6 and β copies of S3 if and only if 6α + 3β = n(n − 1) 2 β ≠ 1, 2 when n is odd, and β ≥ ⌈n/4⌉ when n is even.
In the isogeny-based track of post-quantum cryptography, optimal instances of the signature scheme SQISign rely on primes p such that p ± 1 is smooth. In 2021, a new approach to find those numbers was discovered using solutions to the Prouhet-Tarry-Escott (PTE) problem. With these solutions, we can sieve for smooth integers A and B with a difference of |A − B| = C fixed by the solution. Then some 2A/C and 2B/C are smooth integers hopefully enclosing a prime. They took many different PTE solutions and combined them into a tree to process them more efficiently. But for larger numbers, there are fewer promising PTE solutions, so their advantage over the naive approach, namely checking a single solution at a time, fades. For a single PTE solution, the search can be optimized for the corresponding C and allows one to check smoothness only for those integers that are divisible by C. In this work, we investigate such optimisations and show a significant speed-up compared to the naive approach, both heuristically and empirically. Along the way, we compute the number of roots of a given polynomial modulo prime powers and give an upper bound for the number of roots modulo a composite number.
Rank-metric codes were studied by E. Gabidulin in 1985 after a brief introduction by Delsarte in 1978 as an equivalent of Reed-Solomon codes, but based on linearized polynomials. They have found applications in many areas, including linear network coding and space-time coding. They are also used in cryptography to reduce the size of the keys compared to Hamming metric codes at the same level of security. However, some families of rank-metric codes suffer from structural attacks due to the strong algebraic structure from which they are defined. It therefore becomes interesting to find new code families in order to address these questions in the landscape of rank-metric codes. \par In this paper, we provide a generalization of Subspace Subcodes in Rank metric introduced by Gabidulin and Loidreau. We also characterize this family by giving an algorithm which allows to have its generator and parity-check matrices based on the associated extended codes. We have also studied the specific case of Gabidulin codes whose underlying decoding algorithms are known. Bounds for the cardinalities of these codes, both in the general case and in the case of Gabidulin codes, are also provided.
It is shown in this paper that, if $R$ is a Frobenius ring, then the quaternion ring $\mathcal{H}_{a,b}(R)$ is a Frobenius ring for all units $a,b \in R$. In particular, if $q$ is an odd prime power then $\mathcal{H}_{a,b}(\mathbb{F}_q)$ is the semisimple non-commutative matrix ring $M_2(\mathbb{F}_q)$. Consequently, a homogeneous weight that depends on the field size $q$ is obtained. On the other hand, the homogeneous weight of a finite Frobenius ring with a unique minimal ideal is derived in terms of the size of the ideal. This is illustrated by the quaternions over the Galois ring $GR(2^r,m)$. Finally, one-sided linear block codes over the quaternions over Galois rings are constructed, and certain bounds on the homogeneous distance of the images of these codes are proved. These bounds are based on the Hamming distance of the quaternion code and the parameters of the Galois ring. Good examples of one-sided rate-2/6, 3-quasi-cyclic quaternion codes and their images are generated. One of these codes meets the Singleton bound and is therefore a maximum distance separable code.
In additive combinatorics, a family of finite sets $A_i$ is said to have bounded doubling if there exists a uniform constant $K$ such that $|A_i + A_i|< K|A_i|$ for all i. In this paper, we study such families in the context of certain symmetric Toeplitz matrices over a field F. In particular, we show that if each matrix has bandwidth b and diagonal entries chosen from a finite set $S \subset F$, then the resulting family admits a doubling constant that depends only on b and the additive properties of $S$, but is independent of the matrix dimension. Also, if the diagonals lie in the image of a fixed-dimensional linear map $L: F^m \to F^{b+1}$, then the doubling constant depends on m rather than b. We include examples to illustrate how one-dimensional constraints on S lead to especially small doubling constants.
We construct all nonequivalent (v,k,1) cyclic difference families for 18 sets of parameters v and k for which classification results were not known. We also present the multipliers of all previously classified CDFs with small parameters. Most of the results are double-checked by two different backtrack search algorithms. The usage of an interesting property of the considered objects makes one of these algorithms faster than the other.
Let $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$ and minimum Hamming distance $d$ over $GF(q)$. One of the most important problems in coding theory is to construct codes with best possible minimum distances. In this paper 36 new cyclic and quasi-cyclic (QC) codes over GF(11) are presented and the table from [4] is enlarged by adding three new dimensions.
The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of a finite non-abelian group $H$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $H$ and two vertices, $a^H$ and $b^H$ are adjacent if $a'b' \ne b'a'$ for all $a' \in a^H$ and $b' \in b^H$. In this paper, we compute distance spectrum, distance Laplacian spectrum, distance signless Laplacian spectrum along with their respective energies and Wiener index of NCCC-graphs of $H$ when the central quotient of $H$ is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ (for any prime $p$) or $D_{2n}$ (for any integer $n \geq 3$). As a consequence, we compute various distance spectra, energies and Wiener index of NCCC-graphs of the dihedral group, dicyclic group, semidihedral group along with the groups $U_{(n,m)}$, $U_{6n}$ and $V_{8n}$. Thus we obtain sequences of positive integers that can be realized as Wiener index of NCCC-graphs of certain groups. In particular, we solve Inverse Wiener index Problem for NCCC-graphs of groups when $n$ is a perfect square. We further characterize the above-mentioned groups such that their NCCC-graphs are D-integral, DL-integral and DQ-integral. We also compare various distance energies of NCCC-graphs of the above mentioned groups and characterize those groups subject to the inequalities involving various distance energies.
The concept of a supercharacter theory for a finite group was introduced in 2008 by Diaconis and Iasaacs in [6]. In their article the notion of irreducible characters and conjugacy classes is generalized to superchacters and superclasses while still maintaining important information about the group. This article continues an investigation of a specific supercharacter theory where the supercharacters are taken to be sums of irreducible characters of the same degree. We show this supercharacter theory construction can be done for all projective special linear groups PSL(2,q) and all special orthogonal groups SO(3,q) where q is any power of an (even or odd) prime.
A recursive method is developed for counting domino tilings of a rectangular chessboard (the dimer problem). Based on this method, a new and enhanced recursive algorithm is proposed for solving this problem. Close connections with Fibonacci numbers are traced out.
The Albertson irregularity measure is defined as $Alb(\Gamma)=\sum_{uv\in E(\Gamma)} \vert d(u)-d(v)\vert.$ In this work, the concept of Albertson energy is extended from simple graphs to graphs with self-loops. Also the expression for the Albertson eigenvalues of a graph with self-loops are given. Some bounds on the Albertson energy of graphs with self-loops and the spread of $Alb(\Gamma_S)$ are obtained. In the last section, the Albertson energy of complete, complete bipartite, crown and thorn graphs with self-loops are computed.
In this study, we assume that R is a commutative Noetherian ring with nonzero identity. We present upper bounds for the injective dimension of I, where I is any ideal in the ring R, in terms of the injective dimension of its local cohomology modules and an upper bound for the injective dimension that involves the theory of local cohomology modules. Since I is an ideal in R, we obtain applications of the theory in a general context.
For a left module $_{R}M$ over a non-commutative ring $R$, we define the concept of a strongly semicommutative module as a generalization of the reduced module. This notion constitutes a distinct and stronger category within the class of semicommutative modules. We demonstrate that a module $_{R}M$ is strongly semicommutative if and only if $_{A_{n}(R)}A_{n}(M)$ is strongly semicommutative. Additionally, we establish that $_{R}M$ is strongly semicommutative if and only if $_{R[x]}M[x]$ is strongly semicommutative; this is also equivalent to $_{R[x, x^{-1}]}M[x, x^{-1}]$ being strongly semicommutative. Among our findings, we prove that if $_{R}M$ is strongly semicommutative, then for any reduced submodule $N$ of $M$, the quotient module $M/N$ is also strongly semicommutative. We provide examples of semicommutative modules that are not strongly semicommutative and show that the class of strongly semicommutative modules remains closed under localization.
Let £ be a bounded distributive lattice and S a join closed subset of £. Following the concept of weakly S-2-absorbing submodules, we define weakly S-2-absorbing filters of £. We will make an intensive investigate the basic properties and possible structures of these filters.
