A subset [Formula: see text] of the vertex set of a graph [Formula: see text] is called a perfect code of [Formula: see text] if every vertex of [Formula: see text] is at a distance of no more than one vertex in [Formula: see text]. The biclique partition number of a graph [Formula: see text] is the minimum number of complete bipartite subgraphs (bicliques) required to partition the edge set of [Formula: see text]. The decision form of the minimum biclique cover problem is classified as NP-Complete. Let [Formula: see text] be a finite abelian group and [Formula: see text] be any subset of [Formula: see text]. The Cayley sum graph [Formula: see text] is a simple graph with [Formula: see text] as its vertex set, and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In this paper, we give some conditions on subset [Formula: see text] of [Formula: see text] for finding the perfect code set and inequality of biclique partition number of Cayley sum graph [Formula: see text] and Cayley sum signed graph [Formula: see text].

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets \(\sigma,\rho\) of non-negative integers, a \((\sigma,\rho)\) -set of a graph G is a set S of vertices such that \(|N(u)\cap S|\in\sigma\) for every \(u\in S\) , and \(|N(\!\textit{v})\cap S|\in\rho\) for every \(\textit{v}\not\in S\) . The problem of finding a \((\sigma,\rho)\) -set (of a certain size) unifies standard problems, such as Independent Set , Dominating Set , Independent Dominating Set , and many others. For all pairs of finite or cofinite sets \((\sigma,\rho)\) , we determine (under standard complexity assumptions) the best possible value \(c_{\sigma,\rho}\) such that there is an algorithm that counts \((\sigma,\rho)\) -sets in time \(c_{\sigma,\rho}^{\textsf{tw}}\cdot n^{O(1)}\) (if a tree decomposition of width \(\textsf{tw}\) is given in the input). Let \(s_{{\rm top}}\) denote the largest element of \(\sigma\) if \(\sigma\) is finite, or the largest missing integer \(+1\) if \(\sigma\) is cofinite; \(r_{{\rm top}}\) is defined analogously for \(\rho\) . Surprisingly, \(c_{\sigma,\rho}\) is often significantly smaller than the natural bound \(s_{{\rm top}}+r_{{\rm top}}+2\) achieved by existing algorithms. Toward defining \(c_{\sigma,\rho}\) , we say that \((\sigma,\rho)\) is \({\mathrm{m}}\) -structured if there is a pair \((\alpha,\beta)\) such that every integer in \(\sigma\) equals \(\alpha\) mod \({\mathrm{m}}\) , and every integer in \(\rho\) equals \(\beta\) mod \({\mathrm{m}}\) . Then, setting — \(c_{\sigma,\rho}=s_{{\rm top}}+r_{{\rm top}}+2\) if \((\sigma,\rho)\) is not \({\mathrm{m}}\) -structured for any \({\mathrm{m}}\geq 2\) , — \(c_{\sigma,\rho}=\max\{s_{{\rm top}},r_{{\rm top}}\}+2\) if \((\sigma,\rho)\) is 2-structured, but not \({\mathrm{m}}\) -structured for any \({\mathrm{m}}\geq 3\) , and \(s_{{\rm top}}=r_{{\rm top}}\) is even, and we count the number of edges between — \(c_{\sigma,\rho}=\max\{s_{{\rm top}},r_{{\rm top}}\}+1\) , otherwise, we provide algorithms counting \((\sigma,\rho)\) -sets in time \(c_{\sigma,\rho}^{\textsf{tw}}\cdot n^{O(1)}\) . For example, for the Exact Independent Dominating Set problem (also known as Perfect Code ) corresponding to \(\sigma=\{0\}\) and \(\rho=\{1\}\) , this improves the \(3^{\textsf{tw}}\cdot n^{O(1)}\) algorithm of van Rooij to \(2^{\textsf{tw}}\cdot n^{O(1)}\) . Despite the unusually delicate definition of \(c_{\sigma,\rho}\) , an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair \((\sigma,\rho)\) of finite or cofinite sets where the problem is non-trivial, and any \(\varepsilon > 0\) , a \((c_{\sigma,\rho}-\varepsilon)^{\textsf{tw}}\cdot n^{O(1)}\) -algorithm counting the number of \((\sigma,\rho)\) -sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets \(\sigma\) and \(\rho\) , these lower bounds also extend to the decision version, and hence, our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.

ObjectiveThe ability to reproduce the work of others is an essential part of the scientific disciplines. Replicating observational studies using electronic health record (EHR) data can be challenging due to complexities in data access, variations in EHR systems across institutions, and the potential for unaccounted confounding variables. Our aim is to identify the barriers to methods reproducibility for replication studies using EHR data.MethodsWe replicated a study that examined the risk of hospitalisation following a positive COVID-19 test in individuals with diabetes. Using EHR data from the NHS England’s Secure Data Environment (SDE) covering the whole of England, UK (population 57m), we sought to replicate findings from the original study, which used data from Greater Manchester (a large urban region in the UK, population 2.9m). Both analyses were conducted in Trusted Research Environments (TREs) or SDEs, containing linked primary and secondary care data, however methods reproducibility was not straightforward. Differences between the environments that contributed to the difficulties were documented, categorized into themes, and converted into a list of recommendations for TRE/SDEs.ResultsSmall differences between the environments and the data sources led to several challenges in methods reproducibility. Our recommendations of TRE/SDEs should facilitate future replication studies. The recommendations include: a need for improved machine-readable metadata for EHR data; standardization of governance processes to facilitate federated analysis; mandating of code sharing; and for environments to have a support structure for data engineers and analysts. We also propose a new theme for research, “data reproducibility”, as the ability to prepare, extract and clean data from a different database for a replication study.ConclusionEven with perfect code sharing, data reproducibility remains a challenge. Our recommendations have the potential to reduce the barriers to replication studies and therefore enhance the potential of observational studies using EHR data.

Subgroup perfect codes in lie type simple groups of rank one

<p>Perfect codes in the <span class="math inline">\(n\)</span>-dimensional grid <span class="math inline">\(\Lambda_n\)</span> of the lattice <span class="math inline">\(\mathbb{Z}^n\)</span> (<span class="math inline">\(0<n\in\mathbb{Z}\)</span>) and its quotient toroidal grids were obtained via the truncated distance in <span class="math inline">\(\mathbb{Z}^n\)</span> given between <span class="math inline">\(u=(u_1,\cdots,u_n)\)</span> and <span class="math inline">\(v=(v_1, \ldots,v_n)\)</span> as the graph distance <span class="math inline">\(h(u,v)\)</span> in <span class="math inline">\(\Lambda_n\)</span>, if <span class="math inline">\(|u_i-v_i|\le 1\)</span>, for all <span class="math inline">\(i\in\{1, \ldots,n\}\)</span>, and as <span class="math inline">\(n+1\)</span>, otherwise. Such codes are extended to superlattice graphs <span class="math inline">\(\Gamma_n\)</span> obtained by glueing ternary <span class="math inline">\(n\)</span>-cubes along their codimension 1 ternary subcubes in such a way that each binary <span class="math inline">\(n\)</span>-subcube is contained in a unique maximal lattice of <span class="math inline">\(\Gamma_n\)</span>. The existence of an infinite number of isolated perfect truncated-metric codes of radius 2 in <span class="math inline">\(\Gamma_n\)</span> for <span class="math inline">\(n=2\)</span> is ascertained, leading to conjecture such existence for <span class="math inline">\(n>2\)</span> with radius <span class="math inline">\(n\)</span>.</p>

This paper examines the problem of automated testing of modified programming tasks for future computer science teachers. It is recommended that GitHub Copilot be used to generate tests based on the code. This approach makes it possible to solve the following tasks: reducing the time and effort required for manual checking of programming tasks completed by students; promoting better assimilation of the material of the relevant disciplines by students; promoting the development and improvement of students' skills in algorithmisation and programming; compliance by students with academic integrity; effective use of GitHub Copilot to generate baseline tests to test modified programming assignments completed by students; ensuring the flexibility and scalability of the approach to the development of various training courses in programming; development of students' software product testing skills. In the process of research, we found the following disadvantages of using the GitHub Copilot system for generating basic tests: GitHub Copilot does not always generate perfect code or tests; for complex tasks, GitHub Copilot may require additional correction of the generated code. Therefore, it is important to check and refine the generated tests carefully, if necessary. Therefore, at the moment, we recommend using GitHub Copilot as a template generator for writing tests. The proposed approach is a promising solution for facilitating the verification of modified programming tasks and increasing the effectiveness of the education process of future informatics teachers. The conducted research opens up new prospects for effective improvement of the verification of modified tasks performed by students and the generation of tests for verification. In particular, the integration of the proposed system based on GitHub Copilot with learning management systems (LMS) and automated task verification systems. Another area of research could be exploring the possibilities of using other tools for generating tests instead of GitHub Copilot or combining them in order to obtain better results.

The Use of Genuine Two-Phase Perfect Code for Increased Fiber Sensing Performance

Subgroup Perfect Codes of 2-Groups with Cyclic Maximal Subgroups

Perfect Roman Dominating Functions and Unique Response Roman Dominating Functions are two ways to translate perfect code into the framework of Roman Dominating Functions. We also consider the enumeration of minimal Perfect Roman Dominating Functions and show a tight relation to minimal Roman Dominating Functions. Furthermore, we consider the complexity of the underlying decision problems Perfect Roman Domination and Unique Response Roman Domination on special graph classes. For instance, split graphs are the first graph class for which Unique Response Roman Domination is polynomial-time solvable, while Perfect Roman Domination is NP-complete. Beyond this, we give polynomial-time algorithms for Perfect Roman Domination on interval graphs and for both decision problems on cobipartite graphs. However, both problems are NP-complete on chordal bipartite graphs. We show that both problems are W[1]-complete if parameterized by solution size and FPT if parameterized by the dual parameter or by clique width.

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when [Formula: see text] is an independent set and every vertex of [Formula: see text] is adjacent to exactly one vertex in [Formula: see text]. Let [Formula: see text] = Cay(G, S) be a Cayley graph of a finite group [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when there exists a Cayley graph [Formula: see text] of [Formula: see text] such that [Formula: see text] is a perfect code of [Formula: see text]. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial subgroup perfect code are characterized. In this paper, we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain.

In this paper, we establish the Singleton bound for pomset block codes ([Formula: see text]-codes) of length [Formula: see text] over the ring [Formula: see text]. An upper and a lower bound on the minimum distance of [Formula: see text]-Maximum distance separable (MDS) codes are derived. Moreover, we prove that an MDS [Formula: see text]-code is necessarily an MDS [Formula: see text]-code. We also investigate the relationship between the [Formula: see text]-perfect codes and [Formula: see text]-perfect codes. Given an ideal with partial count and full count, we investigate how MDS and [Formula: see text]-perfect codes relate to one another. Duality theorem is derived for an MDS [Formula: see text]-code when all the blocks are of same length, and the distribution of codewords among [Formula: see text]-balls is analyzed as well. Finally, for a chain pomset, we study the maximum distance separability, packing radius and minimum distance of pomset block codes.

On extended 1-perfect bitrades

Perfect codes in 2-valent Cayley digraphs on abelian groups

Islam teaches Muslims to be kind and tolerant towards non-Muslims, and it does not allow any individual to mistreat even a single person on the basis of differences in beliefs and opinions. The teachings of the Quran and Hadith provide basic guidance to build relationships on the basis of Islamic ethical considerations. The holy Prophet himself established excellent relations with non-Muslims in the fields of politics, society, and economy. Prophet Muhammad benefitted from non-Muslim scholars without any interference from religiously biased prejudice, and also after the holy Prophet, the Four pious caliphs built healthy relationships with non-Muslims’ competent scholars with equal zest. Islam is the well-wisher of the world and teaches humanity to behave with love and affection without any impurities of ill-fated, chaotic, and destructive behaviors towards humanity because Islam is a perfect and complete code of life and works for the betterment of human kind. The lives of the companions of the Prophet Muhammad are beacons for our lives to nurture. Let us have the life of Omar Farooq, whose actions and acts reminded us of lessons of ascension for humans who got destroyed from guidance and destination. His life provides us with codes of conduct to remind us of the adjuration of ascension for humanity. Your luminous examples in decision-making, conquests, and justice provide solutions to every problem for man. That is why a brave new world demands a charter compiled on the teachings and decisions of the caliphate and non-Muslims’ impassionate tutelage, which provides a pivotal point to discuss interdisciplinary and comparative religious problems to solve human problems and inculcate religious harmony to nurture and flourish human society. Keywords: humanity, guidance, ascension, decision, religious

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes C of packing radius r in Z n for r ≥ 2 and n ≥ 3. Recently, Leung and the second author proved this conjecture for linear Lee codes with r = 2. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius 2. We show that if such codes exist in Z n , then n must be

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.

Subgroup total perfect codes in Cayley sum graphs

<p>A perfect directed code (or an efficient twin domination) of a digraph is a vertex subset where every other vertex in the digraph has a unique in- and a unique out-neighbor in the subset. In this paper, we show that a digraph covers a complete digraph if and only if the vertex set of this digraph can be partitioned into perfect directed codes. Equivalent conditions for subsets in Cayley digraphs to be perfect directed codes are given. Especially, equivalent conditions for normal subsets, normal subgroups, and subgroups to be perfect directed codes in Cayley digraphs are given. Moreover, we show that every subgroup of a finite group is a perfect directed code for a transversal Cayley digraph.</p>

Let G be a finite group with the identity element e. The proper order graph of G, denoted by S * (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of S * (G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups.

In this paper, we focus on the perfect and total perfect codes of the non-coprime and coprime graphs associated to the dihedral groups and finite Abelian groups. We used the advantage of independent sets and tried to present the independent polynomial for them.