On the weights of affine-variety codes and some Hermitian codes
Marco Pellegrini, Chiara Marcolla, Massimiliano Sala · 2011
Abstract. For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We use our ideal and a geometric characterization to determine the number of small-weight codewords for some families of Hermitian codes over any Fq. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with d ≤ q. For such codes we also count some other small-weight codewords. Keywords: Affine-variety codes, linear code, distance, minimum-weight word, Hermitian code. 1 Preliminary results Let Fq be a finite field. Let k ≥ 1. For any ideal I in a polynomial ring Fq[X], where X = {x1,..., xk}, we denote by V(I) ⊂ (Fq) k its variety. For any Z ⊂ (Fq) k we denote by I(Z) ⊂ Fq[X] the vanishing ideal of Z. Let g1,..., gs ∈ Fq[X], we denote by I = 〈g1,..., gs 〉 the ideal generated by the gi’s. Let {x q 1 − x1,..., x q k − xk} ⊂ I. Then I is zero-dimensional and radical. Let V(I) = {P1, P2,..., Pn}. We have an isomorphism of Fq vector spaces (an evaluation map): φ: R = Fq[x1,..., xk]/I − → (Fq) n f ↦− → (f(P1),..., f(Pn)). Let L ⊆ R be an Fq vector subspace of R with dimension r. Definition 1. The affine–variety code C(I, L) is the image φ(L) and the affine–variety code C ⊥ (I, L) is its dual code. Our definition is slightly different with respect to that in [1]. Let L be linearly generated by b1,..., br then the matrix