Characterization of Two Classes of Codes that Attain the Griesmer Bound

Linear binary codes are considered. It is shown that, to within isomorphism, there exists a unique code with parameters [$2^k-2^{k-i}-3$, $k$, $2^{k-1}-2^{k-i-1}-2$], $2\leq i\leq k-4$, and two nonisomorphic codes with parameters [$2^k-11$, $k$, $2^{k-1}-6$], $k\geq 5$.