Jacobi sums and cyclotomic matrices involving squares over finite fields

Let q=pⁿ be an odd prime power and let Fq be the finite field of q elements. Let Fq× be the group of all multiplicative characters of Fq and let χ be a generator of Fq×. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over Fq. For example, let s₁,s₂,⋯,s(q-1)/2 be the nonzero squares over Fq. For any integer 1≤ r≤ q-2, define the matrix Bq(r):=[χʳ(sᵢ+sⱼ)+χʳ(sᵢ-sⱼ)]1≤ i,j≤ (q-1)/2. We prove that if q≡ 3 4, then $$ (B_q(r))=∏0≤ k≤ (q-3)/2J_q(χ^r,χ²ᵏ)= {cases} (-1)q-3/4{ i}^nG_q(χ^r)q-1/2/√q & if\ r≡ 1 2, G_q(χ^r)q-1/2/q & if\ r≡ 0 2. {cases},$$ where $J_q(χ^r,χ²ᵏ)$ and $G_q(χ^r)$ are the Jacobi sum and the Gauss sum over $F_q$ respectively.