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On low weight codewords of generalized affine and projective Reed-Muller codes
- Source :
- Designs, Codes and Cryptography, Designs, Codes and Cryptography, 2014, 73, pp.271-297, Designs, Codes and Cryptography, Springer Verlag, 2014, pp.271-297
- Publication Year :
- 2014
- Publisher :
- HAL CCSD, 2014.
-
Abstract
- International audience; 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.
- Subjects :
- Hyperplane
Code
Finite field
[MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT]
Next-to-minimal distance
Weight
Generalized Reed–Muller code
Polynomial
[MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT]
Second weight
Projective Reed–Muller code
Hypersurface
Minimal distance
Homogeneous polynomial
Computer Science::Information Theory
Codeword
Subjects
Details
- Language :
- English
- ISSN :
- 09251022 and 15737586
- Database :
- OpenAIRE
- Journal :
- Designs, Codes and Cryptography, Designs, Codes and Cryptography, 2014, 73, pp.271-297, Designs, Codes and Cryptography, Springer Verlag, 2014, pp.271-297
- Accession number :
- edsair.dedup.wf.001..1945b8156706d312ce764deaa6f734cf