A new extension theorem for linear codes

For an [n,k,d]q code with k 3, gcd(d,q)=1, the diversity of is defined as the pair (Φ0,Φ1) with All the diversities for [n,k,d]q codes with k 3, d −2 (mod q) such that Ai=0 for all i 0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q 5. Double extendability is also investigated.