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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.

MSC:
 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)
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References:
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