On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes

09/18/2023

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by   Sudhir R. Ghorpade, et al.

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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.