On the 3-extendability of quaternary linear codes

We consider the extendability of linear codes over F4, the field of order four. Let C be [n,k,d]4 code with d≡1(mod4), k≥3. The weight spectrum modulo 4 (4-WS) of C is defined as the ordered 4-tuple (w0,w1,w2,w3) with w0=13∑4|i>0Ai, wj=13∑i≡j(mod4)Ai for j=1,2,3. We prove that C is 3-extendable if w0+w2=θk−2 and if either (a) w1−w0<4k−2+4−θk−3; (b) w1−w0>10⋅4k−3−θk−3 or (c) (w0,w1)=(θk−3,6⋅4k−3). We also give a sufficient condition for the l-extendability of [n,k,d]4 codes with d≡4−l(mod4), k≥3 for l=1,2,3 when w0+w2=θk−2+2⋅4k−2.