On the Uniqueness Resp. Nonexistence of Certain codes Meeting the Griesmer Bound

It will be shown that the following binary linear codes are unique: (n, k, d) = (2{suk} − 2u, k, 2{suk−1} − 2u−1} 1 ⩽ u ⩽ k − 1, (2{suk} −2{suk−2} − 3, k,2{suk−1} − 2{suk−3} − 2), k ⩾ 6, and (2{suk−1} + k, k, 2{suk−2} + 2), k ⩾ 3, k ≠ 5. Also there are exactly 2 non-isomorphic (21, 5, 10) codes. Using these results the non-existence of binary linear (n, k, d) codes meeting the Griesmer bound is proved for 2{suk−2} + 3 ⩽ d ⩽ 2{suk−1} − 2{suk−3} − 4.