The weight spectrum of two families of Reed-Muller codes

We determine the weight spectra of the Reed-Muller codes RM(m- 3, m) for m > 6 and RM(m -4,m) for m > 8. The technique used is induction on m, using that the sum of two weights in RM(r -1,m - 1) is a weight in RM(r, m), and using the characterization by Kasami and Tokura of the weights in RM(r, m) that lie between its minimum distance 2m-r and the double of this minimum distance. We also derive the weights of RM(3, 8), RM(4, 9), by the same technique. We conclude with a conjecture on the weights of RM(m - c, m), where c is fixed and m is large enough. & COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper determines the weight spectra of the Reed-Muller codes RM(m-3, m) for m > 6 and RM(m-4, m) for m > 8. The method used is induction on m, utilizing the property that the sum of two weights in RM(r-1, m-1) is a weight in RM(r, m), and the characterization by Kasami and Tokura of weights in RM(r, m) between its minimum distance 2m-r and double this minimum distance. The weights of RM(3, 8) and RM(4, 9) are also derived using the same technique. The paper concludes with a conjecture on the weights of RM(m-c, m) for a fixed c and sufficiently large m.