關於對角型同餘方程及其在最佳衝突迴避碼存在性上的應用

Abstract

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Let p be an odd prime, and let H = ⟨-1,2⟩ be the multiplicative subgroup of Z_p^× generated by −1 and 2. Define l₀ = [Z_p^× ∶ H], the index of H in Z_p^×, and suppose that l₀ = 2q, where q is an odd prime. In this paper, we prove that there exists a generator g ∈ Z_p^× such that the diagonal equation 1+gx^(l₀ )=g^2 y^(l₀). has a solution over Zp. This equation plays a crucial role in the construction of optimal conflict-avoiding codes (optimal CACs) of length p and weight 3, which ensure collisionfree communication in time-slotted multiple-access systems. Our proof is based on the properties of Jacobi sums and cyclotomic numbers.