Multi-valued crosscorrelation function for \(m\)-sequences (Q1430629)

Let \(u_i= \text{tr}_1^n(\alpha\alpha^i)\), \(\nu_i= \text{tr}_1^n(b\alpha^{(q^r+1)i})\), \(a,b\in\mathbb F_{p^n}^*\) \(\alpha\) be a primitive element of \(\mathbb F_{p^n}\), \(q=p^m\), \(n=em\), \(1\leq r\leq e\), \((r,e)=1\), then the crosscorrelation function of a pair \((u,v)\) of \(m\)-sequences is \(\theta_{u,\nu}(\tau)= \sum_{r=0}^{p^n-2} w^{u_i-\nu_{i+\tau}}\), where \(w\) is a complex primitive \(p\)th root of unity, and \(\text{tr}_m^b(\cdot)\) denotes the trace function of \(\mathbb F_q^e\) over \(\mathbb F_q\). Based on the theory of exponential sums and quadratic forms, the authors determine the crosscorrelation function values of a pair of \(m\)-sequences \(u\) and \(v\). For certain particular values for \(m\) and \(r\), the results obtained by the authors are similar to those of R. Gold and K. Nybery.