On the small-weight codewords of some Hermitian codes

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any Fq2Fq2. From these geometric characterizations, we obtain explicit formulas. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with d≤qd≤q and all second-weight codewords for distance-3,43,4 codes.