On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes
Sudhir R. Ghorpade, Rati Ludhani
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.
Topics & Concepts
MathematicsProjective testDimension (graph theory)Code (set theory)Reed–Muller codeMinimum weightMinimum distanceProjective spaceDiscrete mathematicsCharacterization (materials science)CombinatoricsBlocking setOrder (exchange)Projective geometryPure mathematicsLinear codeCollineationBlock codeDecoding methodsAlgebraic geometryAlgorithmComputer scienceSet (abstract data type)NanotechnologyProgramming languageMaterials scienceFinanceEconomicsCoding theory and cryptographygraph theory and CDMA systemsDNA and Biological Computing