# An Improved Dictatorship Test with Perfect Completeness

## TL;DR

This paper introduces a new $k$-query dictatorship test with perfect completeness and improved soundness bounds, advancing the theoretical understanding of hardness proofs for constraint satisfaction problems.

## Contribution

It presents a novel dictatorship test with perfect completeness and better soundness for specific query counts, improving previous bounds.

## Key findings

- Achieves soundness of (2k + 1)/2^k for k = 2^t - 1, t > 2
- Improves upon previous soundness bounds from (2k + 3)/2^k
- Applicable for dictatorship tests with perfect completeness

## Abstract

A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is said to have {\em perfect completeness} if it accepts any dictator function. The {\em soundness} of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a $k$-query dictatorship test with perfect completeness and soundness $ \frac{2k + 1}{2^k}$, where $k$ is of the form $2^t -1$ for any integer $t > 2$. This improves upon the result of \cite{TY15} which gave a dictatorship test with soundness $ \frac{2k + 3}{2^k}$.