## 摘要

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 g^{2}X^{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(ℓ^{2}) 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.

原文 | 英語 |
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文章編號 | 102298 |

期刊 | Finite Fields and their Applications |

卷 | 92 |

DOIs | |

出版狀態 | 已發佈 - 2023 12月 |

## ASJC Scopus subject areas

- 理論電腦科學
- 代數與數理論
- 一般工程
- 應用數學