The Weight Spectrum of the Reed-Muller Codes RM(m – 5,m)
Claude Carlet · IEEE Transactions on Information Theory · 2023
The weight spectra (i.e. the lists of all possible weights) of the Reed-Muller codesRM(r,m), of length 2mand orderr, are unknown forr∈ {3, … ,m- 5} (andmlarge enough). Those ofRM(m-4,m) andRM(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 ofRM(m-5,m) for everym≥ 10. We proceed by first determining the weights inRM(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 29- 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 distancedand 2.5 timesd, and thanks to the fact the weight spectrum is symmetric with respect to 29. Then we use this particular weight spectrum for determining that ofRM(m-5,m), by an induction onm. We check that a recent conjecture (in which we correct a misprint) on the weight spectrum ofRM(m-c,m) is verified forc= 5, and we study the difficulties of trying to extend the results toc≥ 6.