A Boolean function f : 0 1 n 0 1 is called a dictator if it depends on exactly one variable i.e f ( x 1 x 2 x n ) = x i for some i [ 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 2 k 2 k +1 , where k is of the form 2 t − 1 for any integer 2"> t 2 . This improves upon the result of \cite{TY15} which gave a dictatorship test with soundness 2 k 2 k +3 .