Cross-correlations of quadratic form sequences in odd characteristic
Let \(\mathrm{GF}(q)\) be the finite field with \(q\) elements, \(q=p^e\), \(p\) prime. Let \(T_q^{q^n}(\cdot)\) denote the trace function from \(\mathrm{GF}(q^n)\) to \(\mathrm{GF}(q)\) and note that any \(\mathrm{GF}(q)\) linear function from \(\mathrm{GF}(q^n)\) to \(\mathrm{GF}(q)\) can be written as \(T_q^{q^n} (Ax)\) for some \(A \in \mathrm{GF}(q)\). Let \(\alpha\) be a primitive element of \(\mathrm{GF}(q^n)\) and define the sequence \(\mathbf V\) whose \(j\)th element is \(V_j = T_q^{q^n} ({\alpha}^j)\). More generally define the sequence \(\mathbf U\) whose \(j\)th element is \(U_j = T_q^{q^n} (\gamma {\alpha}^{kj} + \delta {\alpha}^j)\). Let \(r\) be a prime number, not necessarily distinct from \(p\) and let \(\zeta\) be a primitive \(r\)th complex root of unity. For any two periodic sequences \(\mathbf S\) and \(\mathbf T\) with elements from \(\mathrm{GF}(r)\), with period \(N\) define their cross-correlation as \[ \Theta_{S,T} (\tau) = \sum_{i=1}^N {\zeta}^{S_i - T_{i + \tau}} . \] Let \(f\) and \(g\) be nonlinear functions from \(\mathrm{GF}(q)\) to \(\mathrm{GF}(r)\) and define the sequence \(\mathbf T\) with \(j\)th element \(g(V_j)\), referred to as a \textit{geometric sequence}. Similarly define the sequence \(\mathbf S\) whose \(j\)th element is \(f(U_j)\), referred to as a \textit{generalized geometric sequence}. Cross correlation functions of a large class of geometric sequences based on \(m\)-sequences in odd characteristic, with larger linear spans than previously known such sequences, are determined. Such sequences are suitable for application in spread spectrum communication systems.
scientific article; zbMATH DE number 1028920