New sufficient conditions for the extendability of quaternary linear codes

For an [n,k,d]4[n,k,d]4 code CC with d   odd, we define the diversity of CC as the 3-tuple (Φ0,Φ1,Φ2)(Φ0,Φ1,Φ2) withΦ0=13∑4|i,i>0Ai,Φj=13∑i≡−j(mod4)Aifor j=1,2 when d≡1(mod4),Φj=13∑i≡j(mod4)Aifor j=1,2 when d≡3(mod4), where AiAi stands for the number of codewords with weight i  . We prove that an [n,k,d]4[n,k,d]4 code with d   odd, k⩾3k⩾3, is extendable if Φ0+Φ2=θk−2+2×4k−2Φ0+Φ2=θk−2+2×4k−2 or if Φ0=θk−4Φ0=θk−4, where θj=(4j+1−1)/3θj=(4j+1−1)/3. For the case when k=3k=3, we determine all possible diversities and the corresponding spectra, which yield that CC is extendable if (Φ0,Φ1,Φ2)∉{(6,1,3),(6,3,3),(2,3,7)}(Φ0,Φ1,Φ2)∉{(6,1,3),(6,3,3),(2,3,7)}. Geometric necessary and sufficient conditions for the non-extendability of CC when (Φ0,Φ1,Φ2)∈{(6,1,3),(6,3,3),(2,3,7)}(Φ0,Φ1,Φ2)∈{(6,1,3),(6,3,3),(2,3,7)} are also given.