On the Aperiodic Correlation Function of Galois Ring m-Sequences

We define Gauss-like sums over the Galois Ring GR(4, r) and bound them using the Cauchy-Schwarz inequality. These sums are then used to obtain an upper bound on the aperiodic correlation function of quadriphase m-sequences constructed from GR(4, r). Our first bound δ1 has a simple derivation and is better than the previous upper bound of Shanbag et. al. for small values of N. We then make use of a result of Shanbag et. al. to improve our bound which gives rise to a bound δimproved which is better than the bound of Shanbag et. al. These results can be used as a benchmark while searching for the best phases—termed auto-optimal phases—of such quadriphase sequences for use in spread spectrum communication systems. The bounds can also be applied to many other classes of non binary sequences.