On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes
Sudhir R. Ghorpade, Rati Ludhani
TL;DR
This paper provides a new proof for the minimum distance of projective Reed-Muller codes, characterizes their minimum weight codewords, and discusses formulas for their dimension, enhancing understanding of their structure and properties.
Contribution
It offers an alternative proof for the minimum distance, fully characterizes minimum weight codewords, and explores dimension formulas for projective Reed-Muller codes.
Findings
Complete characterization of minimum weight codewords
Formulas for the number of minimum weight codewords
Discussion of various dimension formulas
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.
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Taxonomy
TopicsCoding theory and cryptography · Error Correcting Code Techniques · DNA and Biological Computing