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

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

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Published inKragujevac Journal of Mathematics Vol. 48; no. 3; pp. 345 - 364
Main Authors KUMAR, PAVAN, KHAN, NOOR MOHAMMAD
Format Journal Article
LanguageEnglish
Published 2024
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Summary:Recently, linear codes constructed from defining sets have been studied extensively. For an odd prime p, let Trm e be the trace function from Fpm onto Fpe, where e is a divisor of m. In this paper, for the defining set D = {x ∈ F∗ pm : Trm e (x2 + x) = 0} = {d1,d2,...,dn} (say), we define a pe-ary linear code CD by CD ={cx =Trm e (xd1),Trm e (xd2),...,Trm e (xdn) : x ∈ Fpm} and present three-weight and five-weight linear codes with their weight distributions. We show that each nonzero codeword of CD is minimal for m e ≥ 5 and, thus, such codes are applicable in secret sharing schemes.
ISSN:1450-9628
2406-3045
DOI:10.46793/KgJMat2403.345K