Certain diagonal equations and conflict-avoiding codes of prime lengths

Liang Chung Hsia, Hua Chieh Li, Wei Liang Sun*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g2X+gY+1=0 over the finite field Fp for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field Fq of Fp. We show that for q greater than a lower bound of the order of magnitude O(ℓ2) there exists a generator g of Fq× such that the equation in question is solvable over Fq. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

Original languageEnglish
Article number102298
JournalFinite Fields and their Applications
Volume92
DOIs
Publication statusPublished - 2023 Dec

Keywords

  • Binary protocol sequence
  • Conflict-avoiding code
  • Diagonal equation
  • Fibonacci primitive root
  • Hasse-Weil bound
  • Ramanujan's sum

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Algebra and Number Theory
  • General Engineering
  • Applied Mathematics

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