A binary Reed-Muller code 
is the linear code formed
by evaluating all multilinear polynomials of degree
at most 
over the finite field 
at all points of 
, where 
. Its codewords
have length 
. In particular, 
consists of the evaluation vectors of affine Boolean
functions.
Wang et al. (2026) proved that the code covering radius of 
is 242. This is the greatest Hamming distance
from a nine-variable Boolean function to the
nearest affine Boolean function.
Lou and Wang (2026) determined the complete Hamming weight spectrum of 
for every 
, refuting the Carlet-Solé conjecture
in codimension seven. The absent low weights are 322, 326, 330, and 334, together
with their complements relative to 
. More precisely, the weights
are
 |
where 
.
The spectrum contains 
distinct weights.
Zhao and Wu (2026) proved that
 |
Equivalently, every eight-variable Boolean function is within Hamming distance 56 of some Boolean function of polynomial degree at most 3, and the bound is attained. Their proof uses finite covering and compatibility certificates.
Reed-Muller codes also provide binary linear three-query locally correctable codes (Janzer and Manohar 2026).
: A Counterexample to the Carlet-Solé Conjecture."
27 Oct 2026. 
Is 242." 13 Oct 2026. 
Is 56." 2 Nov 2026.