Certain diagonal equations and conflict-avoiding codes of prime lengths

Liang Chung Hsia; Hua Chieh Li; Wei Liang Sun

doi:10.1016/j.ffa.2023.102298

Certain diagonal equations and conflict-avoiding codes of prime lengths

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

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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^ℓ+gY^ℓ+1=0 over the finite field F_p 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 F_q of F_p. We show that for q greater than a lower bound of the order of magnitude O(ℓ²) there exists a generator g of F_q^× such that the equation in question is solvable over F_q. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

Original language	English
Article number	102298
Journal	Finite Fields and their Applications
Volume	92
DOIs	https://doi.org/10.1016/j.ffa.2023.102298
Publication status	Published - 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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10.1016/j.ffa.2023.102298

Cite this

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title = "Certain diagonal equations and conflict-avoiding codes of prime lengths",

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 g2Xℓ+gYℓ+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.",

keywords = "Binary protocol sequence, Conflict-avoiding code, Diagonal equation, Fibonacci primitive root, Hasse-Weil bound, Ramanujan's sum",

author = "Hsia, {Liang Chung} and Li, {Hua Chieh} and Sun, {Wei Liang}",

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doi = "10.1016/j.ffa.2023.102298",

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T1 - Certain diagonal equations and conflict-avoiding codes of prime lengths

AU - Hsia, Liang Chung

AU - Li, Hua Chieh

AU - Sun, Wei Liang

PY - 2023/12

Y1 - 2023/12

N2 - 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 g2Xℓ+gYℓ+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.

AB - 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 g2Xℓ+gYℓ+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.

KW - Binary protocol sequence

KW - Conflict-avoiding code

KW - Diagonal equation

KW - Fibonacci primitive root

KW - Hasse-Weil bound

KW - Ramanujan's sum

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Certain diagonal equations and conflict-avoiding codes of prime lengths

Abstract

Keywords

ASJC Scopus subject areas

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