  The k-Clique problem is a canonical hard problem in parameterized complexity.
In this paper, we study the parameterized complexity of approximating the
k-Clique problem where an integer k and a graph G on n vertices are given as
input, and the goal is to find a clique of size at least k/F(k) whenever the
graph G has a clique of size k. When such an algorithm runs in time T(k)poly(n)
(i.e., FPT-time) for some computable function T, it is said to be an
F(k)-FPT-approximation algorithm for the k-Clique problem.
  Although, the non-existence of an F(k)-FPT-approximation algorithm for any
computable sublinear function F is known under gap-ETH [Chalermsook et al.,
FOCS 2017], it has remained a long standing open problem to prove the same
inapproximability result under the more standard and weaker assumption,
W[1]$\neq$FPT.
  In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e.,
F(k)=O(1)) FPT-approximation algorithms under W[1]$\neq$FPT. In this paper, we
improve this inapproximability result (under the same assumption) to rule out
every $F(k)=k^{1/H(k)}$ factor FPT-approximation algorithm for any increasing
computable function H (for example $H(k)=\log^\ast k$).
  Our main technical contribution is introducing list decoding of Hadamard
codes over large prime fields into the proof framework of Lin.
