Kragujevac Journal of Mathematics
University of Kragujevac - Faculty of Science

Three-Weight and Five-Weight Linear Codes over Finite Fields

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Authors: P. KUMAR AND N. M. KHAN

DOI: 10.46793/KgJMat2403.345K

Abstract:

Recently, linear codes constructed from defining sets have been studied extensively. For an odd prime p, let Trem be the trace function from ????pm onto ????pe, where e is a divisor of m. In this paper, for the defining set D = {x ∈ ????pm∗ : Trem(x2 + x) = 0} = {d1,d2,…,dn} (say), we define a pe-ary linear code ????D by

and present three-weight and five-weight linear codes with their weight distributions. We show that each nonzero codeword of ????_D is minimal for ≥ 5 and, thus, such codes are applicable in secret sharing schemes.

Keywords:

Linear code, weight distribution, Gauss sum, cyclotomic number, secret sharing.

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