On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes
arXiv:2309.10196 · doi:10.3934/amc.2023035
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.
24 pages; to appear in Adv. Math. Commun.; some typos corrected and a reference added in this version