A Study of Fundamental Parameters in Linear Error-Correcting Codes Associated with A(n,k)

Abstract

In this work, q-ary linear codes are constructed from the row space of the |V| x |E| generator matrix of the A(n,k) arrangement graph, where n and k are natural numbers satisfying n > k + 1. The combinatorial structure of arrangement graphs makes them a suitable source for generating well-organized linear codes through their incidence matrices. We investigate the algebraic properties of these codes and determine their key parameters, namely the minimum distance, dimension, and length. These parameters are shown to be [k^2 * Product(i=0 to k)(n-i), Product(i=0 to k-1)(n-i), k(n-k)] over the finite field Fq. Further, we discuss the linear code obtained from the Mobius-Kantor graph, which yields a linear code with parameters [24, 15, 3]. All codes obtained in this manner are linear over the finite field Fq. Each transitive subgroup of the automorphism group provides a permutation decoding set for the associated codes, ensuring that full error correction can be achieved through permutation decoding.