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Holes in Generalized Reed–Muller Codes

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1 Author(s)
Lovett, S. ; Weizmann Inst. of Sci., Rehovot, Israel

The possible relative weights of codewords of Generalized Reed-Muller codes are studied. Let RMq(r,m) denote the code of polynomials over the finite field Fq in m variables of total degree at most r. The relative weight of a codeword f ¿ RMq(r,m) is the fraction of nonzero entries in f. The possible relative weights are studied, when the field Fq and the degree r are fixed, and the number of variables m tends to infinity. It is proved that the set of possible weights is sparse-for any ¿ which is not rational of the form ¿ = ¿/q k, there exists some ¿ > 0 such that no weights fall in the interval (¿-¿,¿+¿). This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.

Published in:

Information Theory, IEEE Transactions on  (Volume:56 ,  Issue: 6 )

Date of Publication:

June 2010

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