Ordered algebraic structure in a linear code

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;

Data

Language:	English
Year of publishing:	2024
Typology:	1.01 - Original Scientific Article
Organization:	UNG - University of Nova Gorica
UDC:	51
COBISS:	217895683
ISSN:	1930-5346
Views:	842
Downloads:	3
Average score:	0 (0 votes)
Metadata:

Other data

Pages:	str. 1248-1259
Volume:	ǂVol. ǂ19
Issue:	ǂissue ǂ4
Chronology:	2025
DOI:	10.3934/amc.2024052
ID:	25459660