Ternary Linear Codes and Quadrics

For an [n, k, d]3 code C with gcd(d, 3) = 1, we define a map wG from Σ = PG(k − 1, 3) to the set of weights of codewords of C through a generator matrix G. A t-flat Π in Σ is called an (i, j)t flat if (i, j) = (|Π ∩ F0|, |Π ∩ F1|), where F0 = {P ∈ Σ | wG(P ) ≡ 0 (mod 3)}, F1 = {P ∈ Σ | wG(P ) 6≡ 0, d (mod 3)}. We give geometric characterizations of (i, j)t flats, which involve quadrics. As an application to the optimal linear codes problem, we prove the non-existence of a [305, 6, 202]3 code, which is a new result.