Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition

1. Introduction

Recent developments in algebra, coding theory, and cryptography have shown that algebraic structures play an important role in both theoretical research and practical applications, with modern communication systems, reliable data transmission, and secure information exchange dependent on efficient coding techniques and strong cryptographic protocols. Many of these techniques are closely related to different branches of algebra, including computational algebra; fuzzy algebraic systems; and semigroup, lattice, and semiring theory.

The aim of this Special Issue is to present recent advances that connect algebraic structures with coding theory and cryptography, with a particular focus on new theoretical results; algorithmic developments; and applications related to encoding and decoding procedures, algebraic methods in coding theory, and mathematical tools used in secure communication systems.

This Special Issue contains eight research papers that present different perspectives on the interactions between algebra and applications in coding theory, cryptography, and computational mathematics.

2. Overview of the Published Papers

Contribution 1 studies radical numerical semigroups. For an integer $n\ge 2$, the radical $r\left(n\right)$ is defined as the product of all positive prime divisors of n. A numerical semigroup S is called a radical if $s+r\left(s\right)\in S$ for every $s\in S\setminus \left\{0\right\}$. The authors present three algorithms that help to analyze the structure of radical semigroups, which make it possible to compute all radical numerical semigroups with a fixed genus, Frobenius number, or multiplicity. Furthermore, the authors prove that for every set X of positive integers with $gcd\left(X\right)=1$, there exists the smallest radical semigroup containing X, and they provide an algorithm to construct it.

Contribution 2 investigates binary cyclically permutable constant-weight codes. The existence problem for optimal $(v,k,1)$ CPCW codes is completely solved for codeword weights $k<5$. The authors consider the smallest open cases, namely $k=5$ and $k=6$, and construct such codes for small values of the code length v and derive necessary conditions for the existence of optimal $\left(k\right(k-1)t+2,k,1)$ CPCW codes. These conditions can be used both to construct new codes and to show that optimal codes with certain parameters do not exist. In particular, they prove that an optimal $(92,6,1)$ CPCW code does not exist.

Contribution 3 focuses on cyclic bases for quotient rings’ modulo zero-dimensional ideals. Based on a necessary and sufficient condition for the existence of cyclic bases, the authors derive an equivalent condition and propose a new algorithm for determining whether a cyclic basis exists, which constructs a matrix and calculates its determinant in order to verify the condition. Compared with the previous approach, the new algorithm reduces the number of candidate elements that must be considered and therefore improves computational efficiency.

In Contribution 4, a new encryption and decryption procedure based on polyadic algebraic structures and signal processing methods is proposed, which uses signals with integer amplitudes to transmit information. Polyadic techniques are applied to convert the plaintext into sequences of special integers. The receiver restores the plaintext by applying specific rules and solving systems of equations. This work demonstrates how algebraic structures can be used to design alternative cryptographic schemes.

Contribution 5 studies a problem related to fractional parts of sequences of the form $\left\{{\xi }_n{\alpha }^n\right\}$. The authors prove that for any sequence of positive numbers ${\xi }_n$ that does not converge to zero faster than an exponential function, and for any sequence of positive numbers ${\delta }_n$, there exists an uncountable set of positive numbers S such that for each $\alpha >1$ in S, there are infinitely many integers n for which the inequality $\left\{{\xi }_n{\alpha }^n\right\}<{\delta }_n$ holds. The result remains valid regardless of how quickly ${\delta }_n$ approaches zero.

Contribution 6 introduces bipolar fuzzy sub-hoops and bipolar fuzzy filters in hoop algebras, structures which extend classical fuzzy logic by allowing both positive and negative membership degrees. The authors study their main algebraic properties and establish several characterizations of bipolar fuzzy filters using level sets. They also determine the conditions under which these filters become implicative filters. These results contribute to the development of fuzzy algebraic systems and their applications in decision-making, image processing, and reasoning under uncertainty.

Contribution 7 introduces layered algebras and applies them to layered graphs, the latter of which provide a useful abstract model for analyzing complex systems, including applications related to coding in communication systems and access control in security. The authors investigate several classes of lattice-based layered algebras and show that the corresponding varieties have decidable equational theories due to the finite model property.

Contribution 8 studies triangular matrices over additively idempotent semirings. The authors determine explicit forms of idempotent and semicentral idempotent triangular matrices, introducing a diamond composition operation and showing that an idempotent $n\times n$ matrix can be represented as the $(n-1)$th power of a sum of diamond compositions of semicentral idempotents. Using this approach, several types of matrices, including strictly upper triangular, unitriangular, and nil-clean matrices, are decomposed into semicentral idempotents.

The papers collected in this Special Issue demonstrate the strong connection between algebraic structures and modern applications in coding theory, cryptography, and computational mathematics, presenting new theoretical results, algorithms, and mathematical techniques that may stimulate further research in these areas.

As the Guest Editor, I would like to thank all of the authors for their valuable contributions to this Special Issue. I also express my sincere gratitude to the reviewers for their careful evaluation and helpful comments, which greatly improved the quality of the published papers. Finally, I would like to thank the editorial office for their support during the preparation of this Special Issue.