Counting subspaces with prescribed Welch sums

Abstract

Let q=2ⁿ with n=2m+1 for an integer m≥1, and let 𝔽_q be the finite field with q elements. Let W_n(x)=x^{2^m+3} be the Welch function on 𝔽_q. For 1≤ k≤ n and b∈𝔽_q, denote by N_n,k(b) the number of k-dimensional 𝔽₂-subspaces E of 𝔽_q such that ∑_{x∈ E}W_n(x)=b. In this paper, we determine N_n,k(b) for all k. The only nontrivial case is k=3, for which the answer is given in terms of a binary Kloosterman sum. As a consequence, we prove a recent conjecture of Hou and Zhao on the third-order sum-freedom of the Welch function.

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BibTeX RIS

BibTeX

@misc{cheng2026counting,
  title = {{Counting subspaces with prescribed Welch sums}},
  author = {Kaimin Cheng and Jie Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15694},
  eprint = {2610.15694},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15694},
}

RIS

TY  - UNPB
TI  - Counting subspaces with prescribed Welch sums
AU  - Kaimin Cheng
AU  - Jie Hu
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15694
UR  - https://arxiv.org/abs/2610.15694
AB  - Let q=2^n with n=2m+1 for an integer m≥1, and let 𝔽_q be the finite field with q elements. Let W_n(x)=x^2^m+3 be the Welch function on 𝔽_q. For 1≤ k≤ n and b∈𝔽_q, denote by N_n,k(b) the number of k-dimensional 𝔽_2-subspaces E of 𝔽_q such that ∑_x∈ EW_n(x)=b. In this paper, we determine N_n,k(b) for all k. The only nontrivial case is k=3, for which the answer is given in terms of a binary Kloosterman sum. As a consequence, we prove a recent conjecture of Hou and Zhao on the third-order sum-freedom of the Welch function.
ER  -

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