Quadratic residue patterns of length 4 and 5

We generalize the results of the first part of the paper by Kiritchenko, Tsfasman, Vl\u{a}du\c{t} and Zakharevich on quadratic residue patterns to the case of arbitrary words over the alphabet $\{R, N\}$ of length $l = 4$ and $l = 5$. First, we obtain explicit formulas for the number of occurrences of a given pattern $S$ in the sequence of quadratic residues and nonresidues modulo a prime $p$. These formulas are expressed in terms of Frobenius traces on a finite collection of low-genus hyperelliptic curves. Second, using a generalized Sato-Tate approach, we describe the limiting distributions of the normalized error term for $l = 4$.