**On a Conjecture of O'Donnell**

*Qichun Wang*

**Abstract: **Let $f:\{-1,1\}^n\rightarrow \{-1,1\}$ be with total degree $d$, and $\widehat{f}(i)$ be the linear Fourier coefficients of $f$. The relationship between the sum of linear coefficients and the total degree is a foundational problem in theoretical computer science. In 2012, O'Donnell Conjectured that
\[
\sum_{i=1}^n \widehat{f}(i)\le d\cdot {d-1 \choose \lfloor\frac{d-1}{2}\rfloor}2^{1-d}.
\]
In this paper, we prove that the conjecture is equivalent to a conjecture on the cryptographic Boolean function. We then prove that the conjecture is true for $d=1,n-1$. Moreover, we count the number of $f$'s such that the upper bound is achieved.

**Category / Keywords: **foundations / Boolean function, Linear coefficient, Total degree, Resiliency

**Date: **received 2 Jan 2020

**Contact author: **qcwang at fudan edu cn

**Available format(s): **PDF | BibTeX Citation

**Version: **20200103:073913 (All versions of this report)

**Short URL: **ia.cr/2020/002

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