Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m)
Abstract
The weight spectra of the Reed-Muller codes RM(r,m) were unknown for r=3,...,m-5. In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of RM(m-5,m) for m10 using the Maiorana-McFarland construction, where the result was tried to be extended to RM(m-6,m), but many problems occurred and much work needed to be done. In this paper, we propose a novel way of constructing Reed--Muller codewords and determine the weight spectrum of RM(m-6,m) for m12, which gives a positive answer to an open question on the weight spectrum of RM(m-c,m) for c=6. Moreover, we put forward a conjecture and verify it for some cases. If the conjecture is true, then that open question can be completely solved.
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