Dual Code Research Articles (Page 3)

Binary and ternary leading-systematic LCD codes from special functions

In this paper, we have introduced a graph $G_{EC}$ generated by type-$(k_{1},k_{2})$ $E$-codes which is $(k_{1},k_{2})$ $E$-torsion graph. The binary code words of the torsion code of $C$ are the set of vertices, and the edges are defined using the construction of $E$-codes. Also, we characterized the graph obtained when $k_{1}=0$ and $k_{2}=0$ and calculated the degrees of every vertex and the number of edges of $G_{EC}$. Moreover, we presented necessary and sufficient conditions for a vertex to be in the center of a graph given the property of the code word corresponding to the vertex. Finally, we represent every quasi-self dual codes of short length by defining the vertex-weighted $(k_{1},k_{2})$ $E$-torsion graph, where the weight of every vertex is the weight of the code word corresponding to the vertex.

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n≤25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl's code table, and some of them are nearly optimal.

Abstract In this paper we use the puncturing and shortening techniques on two already-known classes of optimal cyclic codes in order to obtain three new classes of optimal linear codes achieving the Griesmer bound. The weight distributions for these codes are settled. We also investigate their dual codes and show that they are either optimal or almost optimal with respect to the sphere-packing bound. Moreover, these duals contain classes of almost maximum distance separable codes which are shown to be proper for error detection. Further, some of the obtained optimal linear codes are suitable for constructing secret sharing schemes with nice access structures.

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

In recent time, some mixed types of alphabets have been considered for constructing error correcting codes. These constructions include $\bbbz_{2}\bbbz_{4}-$additive codes, $\bbbz_{2}\bbbz_{2}[u]-$linear codes et cetera. In this paper, we studied a class of codes over a mixed ring $\bbbz_{p}R$ where $R=\bbbz_{p}+v\bbbz_{p}+v^{2}\bbbz_{p}, v^{3}=v.$ We determined an algebraic structure of these codes under certain conditions. We have also constructed a class of LCD cyclic codes over $\bbbz_{p}R$. A necessary and sufficient condition for a cyclic code to be a complementary dual (LCD) code has been obtained.

Linear complementary dual codes have become an interesting sub-family of linear codes over finite fields since they can be practically applied in various fields such as cryptography and quantum error-correction. Recently, properties of complementary dual abelian codes were established in group algebras of arbitrary finite abelian groups. However, the enumeration formulas were given mostly based on number-theoretical characteristic functions. In this article, complementary dual abelian codes determined by some finite abelian groups are revisited. Specifically, the characterization of cyclotomic classes of an abelian group and the enumeration of complementary dual abelian codes are presented, where the group is a finite abelian p-group, a finite abelian 2-group, and a product of a finite abelian p-group and a finite abelian 2-group for some odd prime number p different from the characteristic of the alphabet filed. The enumeration formula for such complementary dual codes is given explicitly in a more precise form without characteristic functions. Some illustrative examples are given as well.

In this study, by using CSS construction, we present the new quantum codes derived from cyclic codes over the commutative non - local ring R = Fp × (Fp + αFp + α2 Fp), where p is an odd prime, α3 = α. Firstly, we investigate the structure of the ring R and study linear codes by defining Lee weight over R. We define a distance – preserving Gray map from R→Fp 4 and discuss the cyclic codes with their dual code by Euclidean inner products on R. Additionally, we find the quantum Singleton defect(QSD) of the quantum code, which is denoted as n + 2 – k – 2d. Magma Software will be employed to provide relevant illustrations for enhanced comprehension.

In this paper, we employ a virus optimization (VO) algorithm, which is one of the metaheuristic optimization techniques, and a known construction method to compute many new binary $ [72,36,12] $ self-dual codes and optimal/near-optimal linear complementary dual (LCD) codes. In particular, we obtain 39 Type Ⅰ and 19 Type Ⅱ codes of length 72, with parameters in their weight enumerators that were not known in the literature before, and 85 new binary LCD codes that are either optimal or near-optimal. We also present the generator matrix of extended Golay code $ [24,12,8] $ by a cyclic group matrix ring element.

The paper focuses on detailing the reasons and effects of code-switching and bilingualism in French Guiana. The primary focus is on analyzing the patterns of code-switching between two dominant languages in the region: The commonly known French Creole and Standard French, which are, furthermore, one of the many, different languages. Using the mixed-methods approach that is based on both quantitative and qualitative methods, the study attempts to find the various social, cultural and linguistic factors that make bilingualism in this very particular sociolinguistic setting feasible. French Guiana being different from other countries as it has different cultures and having different languages coming from different sources. It is the clashing of French Creole and Standard French that points to the history, the social-political dimension and the culture that define the language in this region. Through detailed research of the code-switching in this case study, the aim is to gain a nuanced perception of the intricacies of bilingual communication. The innovative method adapted in the study provides overall understanding of the code-switching patterns applied. Statistical analysis will help us understand code-switching patterns, distribution, while qualitative analysis will give us social and cultural reasons of switching to another language. The effects of the study have made us face the role of language, identity and social conditions. These links affect different disciplines particularly language matters and education. From this study, we will derive such insights that will aid in the making of inclusive policies that will adapt to the diverse linguistic practices. Students attending these schools also benefit from applying the data to their language development programs, those taking into account the multilingual nature of their environments. For example, the study’s conclusion that multilingualism is favorable paves the way for other researchers to come up with more studies on multilingualism. The research undertaken has provided the foundation for further studies to be carried out in relation to the villi context in linguistic studies that will be carried out to identify the same phenomenon in other sociolinguistic contexts. Typically, the outcomes of a this study direct a more general view into dilemmas of dual language and code changing.

For an arbitrary algebrogeometric code and its dual, error-correcting pairs are explicitly calculated. Such a pair consists of codes that are necessary for an efficient decoding algorithm for a given code. The type of pairs depends on the powers of the divisors, with the help of which both the source code and one of the codes included in the pair are constructed. For an algebraic geometric code CL(D, G) of length and associated with a functional field F/Fq of genus g, pairs correcting t = ⌊(n - deg( G) - g - 1)/2⌋ errors, under certain restrictions on the degrees of divisors involved in their construction, are pairs of codes (Cl(D,F),Cl(D,G + F)⊥) or (CL(D,F⊥),Cl(D,F - G)). Restrictions on the degrees of divisors of codes (CL(D,F), CL(D,G - F)) constituting a pair correcting t = ⌊(deg( G) - 3g + 1)/2⌋ errors for the dual code CL(D, G)⊥) Cases where one of the codes involved in constructing a pair belongs to class of MDS codes and the parameters under which this situation is possible are derived. In addition, possible bounds for the divisors involved in the construction of error-correcting pairs for the subfield subcodes CL(D,G)|fp and CL< are calculated /sub>(D, G)⊥)|Fp of the original algebrogeometric code and its dual, with the expansion degree m = 2 (Fq = Fp2).

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

<p>This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field $ \mathbb{F}_q $. In addition, the check matrix of polycyclic codes was provided over $ \mathbb{F}_q $. Specifically, it has been proven that the constacyclic code is an $ {{{\rm{MDS}}}} $ code over $ \mathbb{F}_q $ if and only if its annihilator dual code is also an $ {{{\rm{MDS}}}} $ code. Finally, we have provided some examples of good codes.</p>

Let Z2 = {0, 1}, R1 = Z2 + uZ2, where u2 = 0 and R2=Z2 + uZ2 + vZ2, where u2=v2 = 0=uv=vu. In this article, we study Z2R1R2-additive cyclic, additive constacyclic and additive dual codes and find the structural properties of these codes. The additive cyclic codes are characterized as R2[y]-submodules of a ring S?1,?2,?3 = Z2[y]/?y?1 ? 1? ? R1[y]/?y?2 ? 1? ? R2[y]/?y?3 ? 1?. The extended Gray map is represented by ?1 : Z?1 2 ? R?2 1 ? R?3 2 ?? Z?1+2?2+4?3 2 and is utilized to construct the binary codes with good parameters. The minimal generating polynomials and smallest spanning sets of the above specified codes are obtained. We also establish the relationship between the minimal polynomials of additive cyclic codes and their duals. Further, we provide some examples that support our main results. Finally, the optimal binary codes are determined in Table.

We determine the number of symmetric dualities on the $ s $-fold cross product of the cyclic group of order $ 2, $ which is the additive group of the finite field $ {\mathbb{F}}_{2^s}. $ We show that the ratio of symmetric dualities over all dualities goes to $ 0 $ as $ s $ goes to infinity.We also prove a surprising result that given any two binary codes $ C $ and $ D $ of the same length $ n $ with $ |C||D| = 2^n $, then viewing them as groups there is a symmetric duality $ M $ with $ C^M = D $, which also relates their weight enumerators as additive codes in a group via the MacWilliams relations. Using this theorem we show that any additive code in this setting can be viewed as an additive complementary dual code of length $ 1 $ with respect to some duality.

Let Z2 = {0, 1}, R = Z2 + uZ2, where u2 = 0 and Ruk = Z2 + uZ2 +???+ uk?1Z2, where uk = 0. In this article, we study Z2RRuk-additive cyclic, additive dual codes and their structural properties. The additive cyclic codes are characterized as Ruk [y]-submodules of the ring S?1,?2,?3 = Z2[y]/?y?1 ? 1? ? R[y]/?y?2 ? 1? ? Ruk [y]/?y?3 ? 1?. The extended Gray map is represented by ?1 : Z?1 2? R?2 ? Ruk ?3 ? Z?1,2+2?2+k?3 and is utilized to construct the binary codes with good parameters. The minimal generating polynomials and smallest spanning sets of the above specified codes are obtained. We also establish the relationship between the minimal generating polynomials of additive cyclic codes and their duals. Further, we provide some examples that support our main results. Finally, the optimal binary codes are determined in Table.

As a traditional alcoholic drink and cultural symbol in China, Chinese Baijiu has developed and accumulated for many years, forming many world-class well-known brands such as Maotai, Wuliangye, Luzhou Laojiao and Langjiu. However, with the gradual expansion of Chinese Baijiu consumption industry scale and Chinese Baijiu consumption market, these brand Chinese Baijiu with high brand value and commercial profits were targeted by many counterfeiters, so that a large number of fake and inferior Chinese Baijiu products appeared in the market, disrupting the market. Due to the limitations of the current anti-counterfeiting and traceability technology, it is easy to be imitated and broken through by counterfeiters, making consumers unable to identify the authenticity of Chinese Baijiu products, which seriously infringes the interests of consumers and production enterprises. As a hot topic in many fields, big data can record information from one or more fields, forming a massive digital resource library with high utilization and research value. This project breaks through the traditional technical ideas, changes the single route, and designs and develops a traceability and anti-counterfeiting management system suitable for the field of Chinese Baijiu based on big data, combined with intelligent algorithms, integrating traceability, anti-counterfeiting, sales management and other functions. The key path of this system is one object dual code, digital coding, intelligent algorithms, multi client management, and front-end and back-end separation development.

This theoretical article explores how mental simulation can enhance self-efficacy and stress management in emergency care, focusing on its application in nurse education. This article delves into the concept, functionality, and applications of mental simulation in nurse education, specifically addressing stress mitigation in emergency care. We offer an overview of relevant literature and research, including a specific focus on surgical skills education. The mechanisms of mental simulation, encompassing the simulation theory of action and dual code theory, are discussed, providing valuable insights into neural networks and image evocation processes. The article underscores the significant impact of stress on emergency care performance. It highlights the potential benefits of incorporating mental simulation as a stress-exposure training technique in the learning process. By adopting a Bandurian theoretical framework, the article proposes how mental simulation can contribute to the cultivation of self-efficacious nurses. Integrating mental simulation in nurse education has the potential to mitigate stress’s negative impact on emergency care. Nurses mentally rehearse and simulate scenarios, becoming familiar with demands and stressors. This practice can potentially refine skills, improve performance under pressure, and boost self-efficacy. Mental simulation replicates real-life situations, enhancing learning from a patient safety perspective. Mental simulation enhances self-efficacy and stress management in emergency care. Its integration into nurse education prepares nurses for high-stress situations. Further research should explore its diverse applications. Understanding mental simulation mechanisms can improve training programmes, potentially empowering nurses in delivering self-confident emergency care.

We classify all linear completely regular codes which have covering radiusρ=2and whose dual are antipodal. For this, we firstly show several properties of such dual codes, which are two-weight codes with weightsdandn.

On generalized quasi-cyclic codes over [formula omitted