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On low weight codewords of generalized affine and projective Reed–Muller codes

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Abstract

We propose new results on low weight codewords of affine and projective generalized Reed–Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given.

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  1. Aix-Marseille University, ERISCS and IML case 930, 13288, Marseille Cedex 9, France

    Stéphane Ballet & Robert Rolland

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  1. Stéphane Ballet
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  2. Robert Rolland
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Correspondence to Robert Rolland.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Ballet, S., Rolland, R. On low weight codewords of generalized affine and projective Reed–Muller codes. Des. Codes Cryptogr. 73, 271–297 (2014). https://doi.org/10.1007/s10623-013-9911-7

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