caa a22 4500 475807456 CHVBK 20180406123751.0 cr unu---uuuuu 170329e20000101xx s 000 0 eng 10.1023/A:1008307729806 doi (NATIONALLICENCE)springer-10.1023/A:1008307729806 Vance Todd Department of Defense, 9800 Savage Road, 20755-6000, Fort Meade, MD aut A Gap in GRM Code Weight Distributions [Elektronische Daten] [Todd Vance] The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3m−2 in GRM3(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+Δ, where d is the minimum distance and Δ is the largest integer dividing all weights occurring in the code. Kluwer Academic Publishers, 2000 GRM nationallicence generalized nationallicence Reed-Muller nationallicence code nationallicence weight nationallicence Designs, Codes and Cryptography Kluwer Academic Publishers 19/1(2000-01-01), 27-43 0925-1022 19:1<27 2000 19 10623 https://doi.org/10.1023/A:1008307729806 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1008307729806 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Vance Todd Department of Defense, 9800 Savage Road, 20755-6000, Fort Meade, MD aut NATIONALLICENCE 773 0- Designs, Codes and Cryptography Kluwer Academic Publishers 19/1(2000-01-01), 27-43 0925-1022 19:1<27 2000 19 10623 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer