Some generalizations of extension theorems for linear codes over finite fields

We give four new extension theorems for linear codes over Fq: (a) For q = 2, 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 ≥ 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 is extendable if ∑ i ≡d (mod pt) Ai < q k−1+r(q)qk−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 (θk−1 − 2qk−2, qk−2) is extendable. These are generalizations of some known extension theorems by Hill and Lizak (1995), Simonis (2000) and Maruta (2005).