On the minimum distance of cyclic codes

Abstract

Devirsel kodlarn minimum uzaklıklarn snrlama, kodlama teorisinin klasik problemlerindenbiridir. Wolfmann, iz gösterimleri ve Hilbert 90 Teoremini kullanarak, devirselkodlarn minimum uzaklklar icin baz Artin-Schreier egrilerinin rasyonel noktasaylar cinsinden alt snr buldu. Bu tezde Artin-Schreier egrilerinin denklemleridegistirilerekWolfmann'n snrnn iyilestirilip iyilestirilemeyecegi anlaslmaya calsld.Deneylerimiz iyilestirmenin baz durumlarda mumkun oldugunu gösterdi.Anahtar Kelimeler: Sonlu cisimler, devirsel kodlar, iz gösterimleri, permutasyonpolinomlar

Estimation of the minimum distance of cyclic codes is a classical problem in codingtheory. Using the trace representation of cyclic codes and Hilbert's Theorem 90,Wolfmann found a general estimate for the minimum distance of cyclic codes in termsof the number of rational points on certain Artin-Schreier curves. In this thesis, wetry to understand if Wolfmann's bound can be improved by modifying equations ofthe Artin-Schreier curves by the use of monomial and some nonmonomial permutationpolynomials. Our experiments show that an improvement is possible in some cases.Keywords: Finite elds, cyclic codes, trace representations, permutation polynomials.