Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

  • Authors:

  • Peng Ding;Jennifer D. Key

  • Affiliations:

  • Fair Isaac and Company, Inc., 200 Smith Ranch Road, San Rafael, CA 94903-1996;Department of Mathematical Sciences, Clemson University, Clemson, SC 29634

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2002

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Abstract

We use methods of Mortimer [19] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length \frac{q^m - 1}{q-1} over the finite field Fq of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0rm−1 unless q is prime. The methods of proof are the projective analogue of those developed in [17], and show that the codes spanned by the minimum-weight vectors are spanned over Fq by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.