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Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group H. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some configurations called Brauer configurations. These configurations define some oriented graphs named Brauer quivers which induce a particular class of bound quiver algebras named Brauer configuration algebras. Elements of multisets in Brauer configurations can be seen as letters of the Brauer messages. This paper proves that each point (x,y)∈V=R∖{0}×R∖{0} has an associated Brauer configuration algebra ΛB(x,y) induced by a Brauer configuration B(x,y). Additionally, the Brauer configuration algebras associated with points in a subset of the form (⌊(x)⌋,⌈(x)⌉]×(⌊(y)⌋,⌈(y)⌉]⊂V have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages M(B(x,y)) defined by a semigroup generated by some appropriated matrices A0,A1,A2∈GL(2,R) over a commutative ring R. As an application, we use Brauer messages M(B(x,y)) to construct explicit solutions for systems of linear and nonlinear differential equations of the form X′′(t)+MX(t)=0 and X′(t)−X2(t)N(t)=N(t) for some suitable square matrices, M and N(t). Python routines to compute Cayley hash values of Brauer messages are also included.

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.