Lower Bound for m3(2,37) and Related Code

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Hanan J. AL-Mayyahi, Mohammed A. Alabbood

Abstract

In a finite projective plane PG(2, q), an (k, n)-arc is a set of k points of a projective plane such that some n, but no
n + 1 of them, are collinear. Here, the integer n is the degree of the arc and k ≥ n. The maximum size of an (k,n)-arc in
PG(2,q) is denoted by mn(2,q). In this paper the classification of the (k,3)-arcs in PG(2,37) is presented. It has been obtained
using a computer-based exhaustive search that exploits Secant distributions inequivalent (k,3)-arcs and produces exactly one
representative of each equivalence class. We established that 50 ≤ m3(2,37). The constructed (50,3)-arcs give the respective
lower bounds on m3(2,37). As a consequence there exist new three-dimensional linear codes over GF(37).

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