Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection, error correction, data transmission, and data storage. Codes are studied by various scientific disciplines, such as information theory, electrical engineering, mathematics, linguistics, and computer science, to design efficient and reliable data transmission methods. Many authors in the previous literature have discussed codes over finite fields, Gaussian integers, quaternion integers, etc. In this article, the author defines octonion integers, fundamental theorems related to octonion integers, encoding, and decoding of cyclic codes over the residue class of octonion integers with respect to the octonion Mannheim weight one. The comparison of primes, lengths, cardinality, dimension, and code rate with respect to Quaternion Integers and Octonion Integers will be discussed.

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) $T_1,T_2,\dots ,T_n$. In this case, for $1\le j\le n-1$, two consecutive tiles $T_j$ and $T_{j+1}$ share exactly one edge, either the edge at the east (west) of $T_j$ ($T_{j+1}$) or the edge at the north (south) of $T_j$ ($T_{j+1}$). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras.

Robust data disclosure constitutes an essential problem for contemporary system communication, and the theory of coding becomes the key to maintaining data integrity. This paper discusses constructions and decoding of Bose–Chaudhuri–Hocquenghem (BCH) codes over Quasi-Galois Rings (QGRs) – generalization of classical Galois rings. The QGRs provide richer algebraic structures that lead to improved error correction capabilities, greater codes rates, and more codewords than their Galois counterparts. We provide an all-rounding theory of construction for BCH codes over QGRs, describe them in their construction process, and walk through an efficient decoding technique. Our findings demonstrate the ability of BCH codes under QGR to provide high reliability and performance for the communication systems which will make them a candidate for future use in data transmission and storage.

The research paper examines the design principles and structural features of Generalized Hadamard (GH) codes that operate within Eisenstein local rings Z_(2^s )[ω] utilizing a primitive cube root of unity ω that fulfills ω^2+ω+1=0. The paper first introduces an algebraic Eisenstein integer framework before developing appropriate Gray mapping to examine binary-domain representations of these codes. We establish the essential criteria and necessary checks for determining the linear properties of GH codes based on Z_(2^s ) [ω] structures. This research work defines the kernel structure of these codes together with their rank specification and structural properties evaluation. A classification system for Z_(2^s )[ω]-linear Hadamard codes presents itself in the last part according to their algebraic and combinatorial characteristics. Future studies about coding techniques within algebraic integer rings can begin from our current work because our research expands the understanding of code theory on non-traditional rings.

This paper presents a multiple RGB image encryption scheme that utilizes a pair of 8×8 S-boxes constructed over the residue classes of Eisenstein integers Z[Ω]π, implemented within a three-stage Substitution–Permutation Network (SPN) architecture. The S-boxes are generated using Eisenstein integer algebra through affine transformations and their corresponding inverse functions, ensuring strong nonlinearity. The first S-box serves as a substitution function, while the second contributes to both permutation and diffusion. Enhanced cryptographic strength is achieved through modular arithmetic in Z[Ω]π, which supports essential encryption properties such as confusion and diffusion. Further complexity is introduced by combining the two S-boxesvia an XOR operation to construct a third S-box, promoting greater inter-channel diffusion among the RGB components. The proposed SPN framework is designed to resist differential and linear cryptanalysis through its layered substitution, permutation, and XOR-based mixing operations. Separate yet interlinked processing pathways for each image channel ensure secure and efficient encryption. Experimental evaluations validate the proposed method, demonstrating high entropy, low inter-channel correlation, and robust resistance to various attacks, making it a strong candidate for secure multimedia communication applications.