Classification of the Codewords of Weights 16 and 18 of the Reed-Muller Code RM(n-3,n)

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary
$r^{th}$
-order Reed-Muller code
$RM(r,n)$
can be viewed as the set of all
$n$
-variable Boolean functions of algebraic degree at most
$r$
. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remained the best ones in the literature. The Reed-Muller code
$RM(n-3,n)$
, which corresponds to the dual of the Reed-Muller code
$RM(2,n)$
, has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance
$2^{n-r}$
due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of
${RM}({n-3},{n})$
whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code
${RM}({n-3},{n})$
. We provide a classification of the codewords of weights 16 and 18 of the Reed-Muller code
$RM(n-3,n)$
. Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly.