Search the repository

A+ | A- | | SLO | ENG

Abstract: In this article, we initiate an exploration of the algebraic structures within coding theory. Specifically, we focus on the potential for an ordered al- gebraic structure, known as a BCI-algebra, within an arbitrary linear code C. We demonstrate that any binary linear code C of length n, where n is a positive integer, can be equipped with a BCI-algebra structure between its codewords. This structure is called BCI-algebra over the code C and denoted by (BCI)C -algebra. To establish this structure, we define an operation ∗C be- tween the codewords and investigate its properties. Additionally, we introduce the concept of subcodes within a code and examine the relationship between these subcodes and the ideals of a BCI-algebra over code C. Furthermore, we define a binary relation among codewords and prove that code C, under this relation—referred to as the (BCI)C -order—forms a partially ordered set. Lastly, we show that the generator matrix of a binary linear code C contains the minimal codewords of C with respect to the (BCI)C -order.
Keywords: BCI-algebra, binary linear block codes, subcodes, partially ordered set, lexicographic order
Published in RUNG: 05.12.2024; Views: 1396; Downloads: 3
Link to file
This document has many files! More...

Abstract: Inspired by the concept of BL-algebra as an important part of the ordered algebra, in this paper we investigate the binary block code generated by an arbitrary BL-algebra and study related properties. For this goal, we initiate the study of the BL-function on a nonempty set P based on BL-algebra L, and by using that, l-functions and l-subsets are introduced for the arbitrary element l of a BL-algebra. In addition, by the mean of the l-functions and l-subsets, an equivalence relation on the BL-algebra L is introduced, and using that, the structure of the code generated by an arbitrary BL-algebra is considered. Some related properties (such as the length and the linearity) of the generated code and examples are provided. Moreover, as the main result, we define a new order on the generated code C based on the BL-algebra L, and show that the structures of the BL-algebra with its order and the correspondence generated code with the defined order are the same.
Keywords: BL-function, BL-code, binary linear block codes, coding theory, BL-algebra
Published in RUNG: 07.03.2022; Views: 3041; Downloads: 60
Full text (261,87 KB)