The Weight Spectrum of the Reed-Muller Codes RM(m – 5,m)

The weight spectra (i.e. the lists of all possible weights) of the Reed-Muller codes
$RM(r,m)$
, of length
$2^{m}$
and order
$r$
, are unknown for
$r\in \{3, {\dots },m-5\}$
(and
$m$
large enough). Those of
$RM(m-4,m)$
and
$RM(m-3,m)$
have been determined very recently (but not the weight distributions, giving the number of codewords of each weight, which seem out of reach). We determine the weight spectrum of
$RM(m-5,m)$
for every
$m\ge 10$
. We proceed by first determining the weights in
$RM(5,10)$
. To do this, we construct functions whose weights are in the set
$\{62, 74, 78, 82, 86, 90\}$
, and functions whose weights are all the integers between 94 and
$2^{9}-2=510$
that are congruent with 2 modulo 4 (those weights that are divisible by 4 are easier to determine and they are indeed known). This allows us to determine completely the weight spectrum, thanks to the well-known result due to Kasami, Tokura and Azumi, which precisely determines those codeword weights in Reed-Muller codes which lie between the minimum distance
$d$
and 2.5 times
$d$
, and thanks to the fact the weight spectrum is symmetric with respect to 29. Then we use this particular weight spectrum for determining that of
$RM(m-5,m)$
, by an induction on
$m$
. We check that a recent conjecture (in which we correct a misprint) on the weight spectrum of
$RM(m-c,m)$
is verified for
$c=5$
, and we study the difficulties of trying to extend the results to
$c\geq 6$
.