Weight polarization and divisibility
Discrete Mathematics - Coding Theory
Designs and their codes
The automorphism group of Generalized Reed-Muller codes
Discrete Mathematics
Introduction to Coding Theory
Communication Theory
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The weight distribution of GRM (generalizedReed-Muller) codes is unknown in general. This article describesand applies some new techniques to the codes over \gf{3}.Specifically, we decompose GRM codewords into words from smallercodes and use this decomposition, along with a projective geometrytechnique, to relate weights occurring in one code with weightsoccurring in simpler codes. In doing so, we discover a new gapin the weight distribution of many codes. In particular, we showthere is no word of weight 3^{m-2} in \grm_3(4,m)for m6, and for even-order codes over the ternaryfield, we show that under certain conditions, there is no wordof weight d+\Delta, where d is theminimum distance and \Delta is the largest integerdividing all weights occurring in the code.