On Two Applications of Polynomials xk- cx- d over Finite Fields and More

İrimağzı C., ÖZBUDAK F.

9th International Workshop on the Arithmetic of Finite Fields, WAIFI 2022, Chengdu, China, 29 August - 02 September 2022, vol.13638 LNCS, pp.14-32 identifier

  • Publication Type: Conference Paper / Full Text
  • Volume: 13638 LNCS
  • Doi Number: 10.1007/978-3-031-22944-2_2
  • City: Chengdu
  • Country: China
  • Page Numbers: pp.14-32
  • Keywords: Almost difference families, Golomb costas permutations, Optical CDMA, Optical orthogonal codes, Planar cyclic almost difference sets, Radar, Sonar
  • Middle East Technical University Affiliated: Yes

Abstract

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= 2 n such that q- 1 is a prime or q= 3 n 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 Fq∗ 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 }.