APN Fonksiyonlar ile Tanımlanan Bazı İkili Yarı-mükemmel Lineer Kodlar

Year 2022, Issue: 38, 514 - 519, 31.08.2022

Seher Tutdere

https://doi.org/10.31590/ejosat.1057393

Abstract

2021 yılında, Tutdere, minimum uzaklığı bir tek l sayısından büyük olan bir primitif ikili devirsel kodlar sınıfının örtme yarıçapı R nin r≤R≤l eşitsizliğini sağladığını göstermiştir, burada l, r verilen koda bağlı olan tam sayılardır. Burada, ilk olarak kuadratik APN (hemen hemen mükemmel lineer olmayan) fonksiyon olan Gold fonksiyonlar ile tanımlanan lineer kodların denklikleri incelenmiştir. Daha sonra Tutdere’nin elde ettiği sonucun uygulanarak bu yarı-mükemmel kodların örtme yarıçaplarının hesaplanabileceği gösterilmiştir. 2016 yılında Li ve Helleseth, kuadratik APN fonksiyonlar ile tanımlanan lineer kodların yarı-mükemmel olduklarını göstermişlerdir ve kuadratik olmayan APN fonksiyonlar ile tanımlanan kodların yarı-mükemmel olup olmadığı problemini sunmuşlardır. Burada, sonlu cisim F_(2^m ) , 1≤m≤8 için, üzerinde tanımlanan kuadratik olmayan bazı APN fonksiyonlar ile tanımlanan lineer kodların örtme yarıçapları hesaplanarak bu kodların yarı-mükemmel olduğu gösterilmiştir.

Keywords

APN fonksiyonlar, Sonlu Cisim, Örtme Yarıçapı, Devirli Kodlar.

Some Binary Quasi-perfect Linear Codes Defined by APN Functions

Abstract

In 2021, Tutdere proved that the covering radii R of a class of primitive binary cyclic codes with minimum distance strictly greater than an odd integer l satisfy r≤R≤l, where l, r are some integers depending on the given code. We here first discuss some equivalences of linear codes defined by Gold functions, which are quadratic APN (almost perfect nonlinear) functions. We then show that by applying the result of Tutdere one can find the covering radii of these quasi-perfect codes. In 2016, Li and Helleseth proved that the linear codes defined by the quadratic APN functions are quasi-perfect and they asked whether the linear codes defined by the non-quadratic APN functions are quasi-perfect or not. We here prove that the linear codes defined by some non-quadratic APN functions over the finite field〖 F〗_(2^m ) , for 1≤m≤8, are quasi-perfect, by computing the covering radii of these codes.

Keywords

APN functions, Finite field, Covering radius, Cyclic code

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