Crosscorrelation Properties of Binary Sequences with Ideal Two-Level Autocorrelation

Yu, Nam Yul; Gong, Guang

doi:10.1007/11863854_9

Cited by 9 publications

(12 citation statements)

References 14 publications

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“…Hadamard equivalence for cyclic Paley-Hadamard difference sets was defined in [12] and applications of these equivalences was discussed in [20,21]. These equivalences are essentially some forms of our product property.…”

Section: Remark 23 the Identitymentioning

confidence: 96%

Abelian difference sets of order n dividing λ

Arasu

Chen

Dillon

et al. 2007

Des. Codes Cryptogr.

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Section: Remark 23 the Identitymentioning

confidence: 96%

Abelian difference sets of order n dividing λ

Arasu

Chen

Dillon

et al. 2007

Des. Codes Cryptogr.

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“…For the three-term case, we conjecture that the WG transform of the three-term function is invariant under the WG transform (see Section 3). It is turned out that that the above decimations are the same decimations in [39] where Yu and Gong studied their respective Hadamard transforms of W G T 5 (x d ) and T 3(x d ). Using this result, we are able to establish that the uniformity in binary tuples of various lengths in the filtering sequences.…”

Section: Introductionmentioning

confidence: 93%

“…Hence the result is established. In Section 2.3, we mentioned that there are two ways to define the WG transformations (see Facts 1 and 2 and Fact 3) where the exponents are calculated using either m = 3s ± 1 or 3k ≡ 1 mod m. In [39], Yu and Gong proved the Hadamard transform of the decimated WG transformation (see Fact 5) where the decimation is d 1 = 2 2s −2 s +1 2 s +1 . We provide a connection between the decimations d 1 and d defined by Eq.…”

Section: Invariance Of the Wg Transformationmentioning

confidence: 99%

On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators

Mandal¹,

Gong²

2022

AMC

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Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period <inline-formula><tex-math id="M2">\begin{document}$ 2^n $\end{document}</tex-math></inline-formula> and a filtering function in <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple uniformity in output sequences, up to <inline-formula><tex-math id="M5">\begin{document}$ t = (n-m+1) $\end{document}</tex-math></inline-formula>. The conjecture of Mandal et al. in [<xref ref-type="bibr" rid="b29">29</xref>] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee <inline-formula><tex-math id="M6">\begin{document}$ (n-m+1) $\end{document}</tex-math></inline-formula>-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.

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“…If the maximum crosscorrelation of a pair of binary sequences of period 2 n − 1 is much larger than the optimum value, then the pair is not so attractive in practice. The pairs of binary sequences having correlation with at most 5 possible values have been of significant interest, see a summary of these cases in [58] and references therein. It is generally challenging to settle the correlation distribution for a pair of binary sequences.…”

Section: 3 C R O S S C O R R E L At I O N S B E T W E E N T W O -mentioning

confidence: 99%

“…The correlation distributions for many pairs leading to 3-valued and 4-valued crosscorrelation have been determined [6,11,18,26,31,37,46]. However, it is more difficult to determine the correlation distribution of those pairs leading to 5-valued crosscorrelation and most of the correlation distributions are left open [30,35,36,58].…”

Section: 3 C R O S S C O R R E L At I O N S B E T W E E N T W O -mentioning

confidence: 99%