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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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[1] Baumert, L.D.; McEliece, R.J., Weight of irreducible cyclic codes, Information and control, 20, 158-175, (1972) · Zbl 0239.94007
[2] Carlitz, I.; Uchiyama, S., Bounds for exponential sums, Duke math. J., 24, 37-41, (1957) · Zbl 0088.03901
[3] Gold, R., Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE trans. information theory, 14, 154-156, (1967) · Zbl 0228.62040
[4] Golomb, S.W., Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences, Information sci., 1, 87-109, (1968) · Zbl 0238.20060
[5] Kasami, T., Weight enumerators for several classes of subcodes of the second order binary Reed-muller codes, Information and control, 18, 369-394, (1971) · Zbl 0217.58802
[6] Kasami, T.; Lin, S.; Peterson, W.W., Some results on cyclic codes which are invariant under the affine group and their applications, Information and control, 11, 475-496, (1968) · Zbl 0169.51101
[7] Mattson, H.E.; Solomon, G., A new treatment of Bose-chaudhuri codes, J. SIAM, 9, 654-669, (1961) · Zbl 0137.13604
[8] McEliece, R.J.; Rumsey, H., Euler products, cyclotomy, and coding, J. number theory, 4, 302-311, (1972) · Zbl 0235.12014
[9] McEliece, R.J., Weight congruences for p-ary cylic codes, Discrete math, 3, 177-192, (1972) · Zbl 0251.94008
[10] Niho, Y., Multi-valued cross-correlation function between two maximal linear recursive sequences, ()
[11] Pless, V., Power moment identities on weight distributions in error-correcting codes, Information and control, 6, 147-152, (1963) · Zbl 0149.37905
[12] Storer, T., Cyclotomy and difference sets, (1967), Markham Chicago · Zbl 0157.03301
[13] Trachtenberg, H.M., On the cross-correlation functions of maximal linear sequences, ()
[14] Lint, J.H.Van, Coding theory, (1971), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.