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

AbstractWe 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 0<r<m−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.