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The Cross-Correlation Distribution of the Niho-Type Decimation d=4(2mβˆ’1)+1

  • Maosheng Xiong
  • Haode Yan
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Abstract

The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation d=4(2mβˆ’1)+1 over 𝔽22m for every positive integer m. With q=2m, this is equivalent to determining the distribution of the number of roots in Uq+1 of fa(x)=x7+ax4+aΜ… x3+1 for aβˆˆπ”½q2, where Uq+1={xβˆˆπ”½q2:xq+1=1}. The main difficulty is to count the four-element subsets of Uq+1 that may occur as root sets of fa. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over 𝔽q, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over 𝔽22m and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.

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Cite

BibTeXRIS
BibTeX
@misc{xiong2026crosscorrelation,
  title = {{The Cross-Correlation Distribution of the Niho-Type Decimation $d=4(2^m-1)+1$}},
  author = {Maosheng Xiong and Haode Yan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.04683},
  eprint = {2609.04683},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04683},
}
RIS
TY  - UNPB
TI  - The Cross-Correlation Distribution of the Niho-Type Decimation d=4(2^mβˆ’1)+1
AU  - Maosheng Xiong
AU  - Haode Yan
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.04683
UR  - https://arxiv.org/abs/2609.04683
AB  - The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation d=4(2^mβˆ’1)+1 over 𝔽_2^2m for every positive integer m. With q=2^m, this is equivalent to determining the distribution of the number of roots in U_q+1 of f_a(x)=x^7+ax^4+aΜ… x^3+1 for aβˆˆπ”½_q^2, where U_q+1={xβˆˆπ”½_q^2:x^q+1=1}. The main difficulty is to count the four-element subsets of U_q+1 that may occur as root sets of f_a. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over 𝔽_q, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over 𝔽_2^2m and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.
ER  -