Certain Types of Linear Codes over the Finite Field of Order Twenty-Five

Emad Bakr Al-Zangana,Elaf Abdul Satar Shehab

doi:10.24996/ijs.2021.62.11.22

Emad Bakr Al-Zangana, Elaf Abdul Satar Shehab

Open Access

https://doi.org/10.24996/ijs.2021.62.11.22

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Journal: Iraqi Journal of Science	Publication Date: Nov 30, 2021
License type: cc-by

Affiliation: Mustansiriyah University

Abstract

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of were studied over different finite fields.

Highlights

Let ( )denotes the Galois field of elements, is a prime power, is a plus point at infinity, and is the vector space of row vectors of length with entries in
-MDS and an arc in the projective space, where this equivalence comes from the fact that the matrix in which each column is a point of an arc has formed a generator matrix of
An algorithm was executed with GAP program to compute the generator matrices of linear codes from over several finite fields

Summary

Introduction

Denotes the Galois field of elements, is a prime power, is a plus point at infinity, and is the vector space of row vectors of length with entries in. Let denotes the number of codewords with Hamming weight in a code of length. ) is called the weight distribution of the code. Two linear codes are isomorphic (equivalent) if the generator matrices are equivalent after doing a sequence of row (column) operations. The code is called projective if the columns of a generator matrix are pairwise linearly independent and denoted by. -MDS and an arc in the projective space, where this equivalence comes from the fact that the matrix in which each column is a point of an arc has formed a generator matrix of. Every -set, that is ( )-arc, in ( ) gives a generator matrix of MDS -ary ,. The second objective is to construct linear codes from the incidence matrix of lines and points of. The GAP programming was used to perform the calculations required for achieving the desired results [12]

Non-Isomorphic Error-Correcting

Codes from Incidence Matrix

Conclusions