A Gap in GRM Code Weight Distributions

Author: Vance T.D.

Source: Designs, Codes and Cryptography, Volume 19, Number 1, January 2000 , pp. 27-43(17)

Publisher: Springer

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Abstract:

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 \gf{3}. 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 3^{m-2} in \grm_3(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+\Delta, where d is the minimum distance and \Delta is the largest integer dividing all weights occurring in the code.

Keywords: GRM; generalized; Reed-Muller; code; weight

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Defense, 9800 Savage Road, Fort Meade, MD 20755-6000

Publication date: 2000-01-01