The serial test for a nonlinear pseudorandom number generator

Let M = 2w, and GM = {1, 3, ...,M − 1}. A sequence {yn}, yn ∈ GM , is obtained by the formula yn+1 ≡ ayn + b+ cyn mod M. The sequence {xn}, xn = yn/M, is a sequence of pseudorandom numbers of the maximal period length M/2 if and only if a + c ≡ 1 (mod 4), b ≡ 2 (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.