Codes from the incidence matrix of the essential graph

Darani A. Y., Soheilnia F., Ugurlu E. A.

Italian Journal of Pure and Applied Mathematics, no.52, pp.245-254, 2024 (ESCI, Scopus) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Journal Name: Italian Journal of Pure and Applied Mathematics
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Compendex, zbMATH
  • Page Numbers: pp.245-254
  • Keywords: essential graph, incidence matrix, linear code
  • Marmara University Affiliated: Yes

Abstract

Let R be a commutative ring and M be an R-module. The essential graph over M, denoted by EG(M), is defined as a graph associated to M with vertex set Z(M) \ AnnR(M), and a pair of distinct vertices x and y are adjacent if and only if AnnM(xy) is an essential submodule of M. In this paper, we investigate the linear codes with respect to the Hamming weight from incidence matrix of the essential graphs over M. If Zn be the ring of integer module n, then EG(Zn) is a linear code. Let p1 and p2 be distinct prime numbers. It is shown that if n = p1p2, then C2(EG(Zn)) = ［(p1 − 1)(p2 − 1), p1 + p2 − 2, min{p1 − 1, p2 − 1}］2. Moreover if n = pα11 pα22 with αi ≥ 1 for i = 1, 2, then C2(EG(Zn)) = ［|E|, |V | − 1, min{p1 + 1, p2 + 1}］2