On certain determinants involving squares over finite fields

In this paper, motivated by a recent work of Z.-W. Sun and by using algebraic properties of Jacobi sums and the theory of almost circulant matrices posed by H.-L. Wu and L.-Y. Wang, we determine the explicit value of the determinant of the matrix $$ Y_{q,d}(\psi )=[\psi (s_i+ds_j)+\psi (s_i-ds_j)]_{2\le i,j\le (q-1)/2}, $$ where $q\equiv 3\pmod 4$ is a prime power, $d$ is a non-zero element over the finite field $\mathbb {F}_q$, $\psi $ is a non-trivial multiplicative character of $\mathbb {F}_q$ and $s_1=1,s_2,\ldots ,s_{(q-1)/2}$ are all non-zero squares over $\mathbb {F}_q$.