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Two families of Niho sequences having four-valued cross correlation with \(m\)-sequences. (English) Zbl 1412.11141

Summary: For two odd integers \(m\) and \(s\) with \(1\le s < m\) and \(\gcd(m,s) = 1\), let \(h\) satisfy \(h(2^s - 1)\equiv 1 \pmod{2^m + 1}\) and \(d = (h + 1)(2^m - 1) + 1\). The cross correlation function between a binary \(m\)-sequence of period \(2^{2m} - 1\) and its \(d\)-decimation sequence is proved to take four values, and the correlation distribution is completely determined. Let \(n\) be an even integer and \(k\) be an integer with \(1 \le k \le \frac{n}{2}\). For an odd prime \(p\) and a \(p\)-ary \(m\)-sequence \(\{s(t)\}\) of period \(p^n - 1\), define \(u(t) = \sum_{i=0}^{\frac{p^k - 1}{2}} s(d_it)\), where \(d_i = ip^{\frac{n}{2}} + p^k - i\) and \(i = 0,1,\dots, \frac{p^k - 1}{2}\). It is proved that the cross correlation function between \(\{u(t)\}\) and \(\{s(t)\}\) is three-valued or four-valued depending on whether \(k\) is equal to \(\frac{n}{2}\) or not, and the distribution is also determined.

MSC:

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
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