On two applications of polynomials x^k-cx-d over finite fields and more

Date

2023-01-01

Author

İrimağzı, Canberk
Özbudak, Ferruh

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For integers k∈[2,q−2] coprime to q−1 , we first bound the number of zeroes of the family of polynomials xk−cx−d∈Fq[x] where q=2n such that q−1 is a prime or q=3n such that (q−1)/2 is a prime. This gives us bounds on cross-correlation of a subfamily of Golomb Costas arrays. Next, we show that the zero set of xk−cx−d over Fq is a planar almost difference set in F∗q and hence for some set of pairs (c, d), they produce optical orthogonal codes with λ=1 . More generally, we give an algorithm to produce optical orthogonal codes (OOCs) from P(x)=xℓ1+cℓ2xℓ2+cℓ2−1xℓ2−1+⋯+c1x∈Fq[x] where interestingly ℓ1≫ℓ2 . We focus on the case ℓ2∈{2,3} and provide examples of (q−1,w,λ) -OOCs with λ∈{2,3} .

URI

https://doi.org/10.1007/978-3-031-22944-2_2
https://hdl.handle.net/11511/102381

Relation

Arithmetic of Finite Fields

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Department of Mathematics, Book / Book chapter

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Citation Formats

C. İrimağzı and F. Özbudak, On two applications of polynomials x^k-cx-d over finite fields and more. 2023.