Some generalizations of extension theorems for linear codes over finite fields

We give four new extension theorems for linear codes over F-q: (a) For q = 2(h), h >= 3, every [n, k, d](q) code with d odd whose weights are congruent to 0 or d (mod q/2) is extendable. (b) For q = 2(h), h >= 3, every [n, k, d](q) code with gcd(d, q) = 2 whose weights are congruent to 0 or d (mod q) is doubly extendable. (c) For integers h, m, t with 0 <= m < t <= h and prime p, every [n, k, d](q) code with gcd(d, q) = p(m) and q = p(h) is extendable if Sigma(i not equivalent to d) (mod p(t)) A(i) < q(k-1) + r(q)q(k-3)(q-1), where q+ r(q) + 1 is the smallest size of a non-trivial blocking set in PG(2, q). (d) Every [n, k, d](q) code with gcd(d, q) = 1 whose diversity is (theta(k-1) -2q(k-2), q(k-2)) is extendable. These are generalizations of some known extension theorems by Hill and Lizak (1995), Simonis (2000) and Maruta (2005).