The second weight of generalized Reed-Muller codes in most cases

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d (n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.Comment: This version corrects minor misprints and gives a more detailed proof of a combinatorial lemm

Remarks on low weight codewords of generalized affine and projective Reed-Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.Comment: New version taking into account recent results from Elodie Leducq on the characterization of the next-to-minimal codewords (cf. arXiv:1203.5244