Subcodes of the projective generalized Reed-Muller codes spanned by minimum-weight vectors (Q1611364)

The authors prove, alternative to a dimension argument, that neither the projective generalized Reed-Muller code \(P_{F_q}(r,m)\) of order \(r\) and of length \((q^m-1)/(q-1)\) over a finite field \(F_q\), nor its dual, is spanned by its minimum-weight vectors for \(0<r<m-1\) unless \(q\) is a prime. The minimum-weight codewords of \(P_{F_q}(r,m)\) are scalar multiples of the incidence vectors of the \((m-r-1)\)-dimensional projective subspaces of \(PG(m-1,q)\), and the minimum-weight vectors of \(P_{F_q}(r,m)^{\perp}\) are the scalar multiples of the incidence vectors of the sets \((\pi_1 \cup \pi_2)\setminus (\pi_1 \cap \pi_2)\) where \(\pi_1\) and \(\pi_2\) are two \(r\)-dimensional subspaces of \(PG(m-1,q)\) intersecting in an \((r-1)\)-dimensional space. The methods of the authors also show that the codes spanned by the minimum-weight vectors are spanned over \(F_q\) by monomial functions in the \(m\) variables. The authors end the article by looking at the subfield subcodes of the projective generalized Reed-Muller codes and the duals of these subfield subcodes. It is shown that the dual of the binary subfield subcode of \( P_{F_{2^t}}(1,4)\) has a basis of incidence vectors of hyperovals in \(PG(3,2^t)\), i.e., a basis of minimum-weight vectors.