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Ordered algebraic structure in a linear codeHashem Bordbar, 2024, izvirni znanstveni članek
Opis: 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.
Ključne besede: BCI-algebra, binary linear block codes, subcodes, partially ordered set, lexicographic order
Objavljeno v RUNG: 05.12.2024; Ogledov: 380; Prenosov: 2
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