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Some results about the cross-correlation function between two maximal linear sequences. (English) Zbl 0348.94017
Let \(\{a_j\}\) and \(\{a_{dj}\}\) be two maximal linear sequences of period \(p^n-1\). The cross-correlation function is defined by \[ C_d(t)=\sum_{j=0}^{p^n-2} \zeta^{a_{j-t}-a_{dj}} \qquad \text{for}\;t=0,1,\ldots,p^n-2 \] where \(\zeta=\exp(2\pi i/p)\). Finding the values and the number of occurrences for each value of \(C_d(t)\) is equivalent to finding the complete weight enumerator for the cyclic \((p^n-1,2n)\) code with parity-check polynomial which is the product of the recursion polynomials for the two maximal linear sequences. here properties of \(C_d(t)\) are investigated. An expression for \[ \sum_{t=0}^{p^n-2} C_d(t)C_d(t+\tau_1)\dots C_d(t+\tau_{n-1}) \] is derived. When \(\tau_1=\tau_2=\ldots=\tau_{n-1}=0\) this is an analogue to the Pless power moment identities which is often used in calculation of the Hamming weight enumerator. When \(d\not\equiv p^i\pmod{p^n-1}\) it is shown that \(C_d(t)\) has at least three different values. We also provide an upper bound on the number of different values of \(C_d(t)\) for some choices of \(d\). Further, the values and number of occurrences of each value of \(C_d(t)\) is determined completely for several new decimations \(d\) when \(C_d(t)\) has less than or equal to six different values. Numerical results and some conjectures are given.

94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
94B15 Cyclic codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: DOI
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