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 ∈ {3, … , 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 ≥ 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 wellknown 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 2 ⁹ . 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 ≥ 6.