On the Linear Complexity of Sidel’nikov Sequences over \({\mathbb {F}}_d\)

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We study the linear complexity of sequences over the prime field \({\mathbb{F}_d}\) introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation.

The first author has been supported by the Austrian Science Fund (FWF) grant S83 and by the Austrian Academy of Sciences.