Second weight codewords of generalized Reed-Muller codes

Abstract

Recently, the second weight of generalized Reed-Muller codes have been determined (Erickson 1974; Bruen 2010; Geil, Des. Codes Cryptogr. 48(3):323–330, 2008; Rolland, Cryptogr. Commun. 2(1):19–40, 2010). In this paper, we give the second weight codewords of the generalized Reed-Muller codes.

Appendix: Blocking sets

Blocking sets have been studied by Erickson in [8] in the case of affine planes and by Bruen in [4, 3, 5] in the case of projective planes.

Definition 1

Let S be a subset of the affine space ${\mathbb{F}}_q^2$. We say that S is a blocking set of order n of ${\mathbb{F}}_q^2$ if for all line L in ${\mathbb{F}}_q^2$, #(S ∩ L) ≥ n and $\#(({\mathbb{F}}_q^2\setminus S)\cap L)\geq n$.

Proposition 10

(Lemma 4.2 in [ 8 ]) Let q ≥ 3, 1 ≤ b ≤ q − 1 and f ∈ R _q (b,2). If f has no linear factor and |f| ≤ (q − b + 1)(q − 1), then the support of f is a blocking set of order (q − b) of ${\mathbb{F}}_q^2$ .

In [8] Erickson make the following conjecture. It has been proved by Bruen in [5].

Theorem 13

(Conjecture 4.14 in [ 8 ]) If S is a blocking set of order n in ${\mathbb{F}}_q^2$ , then #S ≥ nq + q − n.