New sufficient conditions for the extendability of quaternary linear codes

For an [n,k,d]4 code with d odd, we define the diversity of as the 3-tuple (Φ0,Φ1,Φ2) with where Ai stands for the number of codewords with weight i. We prove that an [n,k,d]4 code with d odd, k 3, is extendable if Φ0+Φ2=θk−2+2×4k−2 or if Φ0=θk−4, where θj=(4j+1−1)/3. For the case when k=3, we determine all possible diversities and the corresponding spectra, which yield that is extendable if (Φ0,Φ1,Φ2) {(6,1,3),(6,3,3),(2,3,7)}. Geometric necessary and sufficient conditions for the non-extendability of when (Φ0,Φ1,Φ2) {(6,1,3),(6,3,3),(2,3,7)} are also given.