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On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes (2309.10196v2)
Published 18 Sep 2023 in cs.IT, math.CO, and math.IT
Abstract: We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed.