On the Extendability of Linear Codes

R. Hill and P. Lizak (1995, in “Proc. IEEE Int. Symposium on Inform. Theory, Whistler, Canada,” pp. 345) proved that every [n, k, d]q code with gcd(d, q)=1 and with all weights congruent to 0 or d (modulo q) is extendable to an [n+1, k, d+1]q code with all weights congruent to 0 or d+1 (modulo q). We give another elementary geometrical proof of this theorem, which also yields the uniqueness of the extension.