By using finite fields of order \(p^r\) with \(r \ge 2\) and their quadratic characters, Sarkozy and Winterhof presented a construction for binary sequences of length \(p^r\) with strong pseudorandom properties. In the special case \(r=2\) Gyarmati improved on their estimates for the pseudorandom measures of these sequences. Here we extend Gyarmati’s result and sharpen the estimates of Sarkozy and Winterhof for any prime power \(p^r\).