Geometric Construction of a (57, 2)-Blocking Set in PG(2, 19) and Analysis of the [324,3,306]_19Griesmer Code
DOI:
https://doi.org/10.54153/sjpas.2025.v7i1.1010Keywords:
Arc, Bounded Griesmer , duble Blocking group, dropping [n,k,d]_qcode. projective techniques Plane, Optimal Linear codeAbstract
In this paper, we explore the geometric structure of (57, 2)-blocking set in the projective plane PG(2, 19). By leveraging this structure, we construct a new (324, 18)-arc and derive a novel linear code with parameters . Additionally, we systematically analyze the Grismer bound to determine whether this code is optimal or non-optimal, providing rigorous evidence through detailed stratification. Our investigation includes examples of arcs in the finite field PG(2, 19), and we demonstrate how these constructions contribute to the broader understanding of coding theory and finite geometry. The study also introduces new methodologies for identifying and characterizing blocking sets, arcs, and linear codes, expanding the potential for error correction, data transmission, and cryptography applications. By presenting concrete examples and theoretical insights, we aim to bridge the gap between geometric constructions and their practical implications in coding theory. Furthermore, this research underscores the significance of projective geometry in developing innovative solutions to long-standing problems in combinatorics and information theory. Through these findings, we contribute to the ongoing advancement of optimal code discovery and analysis within the finite field context
References
1. J.W.P.Hirschfeld , projective techniques Geometric over finite fields ..(1979)
2. S. Ball, A. Blokhuis, On the size ofa double blocking group in PG(2, q), Finite Fields Appl. 2(1996) 125–137.
3. R.N.Daskalov ,On the maximum size of some (k,r)-arcs in PG(2,q) University of Gabrovo (2008)
4. R.N.Daskalov, New minmum distance bounds for Linear Codes over GF(5) University of Gabrovo (2004).
5. yahya ,N.Y.k. A Geometric Construction of complete(k,r)-arc in PG(2,7) and the Related projective [n,3,d]7 Codes ,Raf.j,of Comp,&Math's.vol. 12,No.1,2018,University of mosul,Iraq
6. R.N.Daskalov,The best Known (k,r)-arcs in PG(2,19) University of Gabrovo (2017)
7. S.Ball , Table on (n,r)-arcs limer codes on Three dimensional www://htpww
8. S.Ball,J.w.p. Hirschfeld ,Bounds on (n,r)-arc and their application to linear codes,Finite Fields Their Appl.11(2005)326-336.
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