Classification and nonexistence results for linear codes with prescribed minimum distances

Starting from a linear [n, k, d] _q code with dual distance ${d^{bot}}$ , we may construct an ${[n - d^bot, k - d^bot +1,geq d]_q}$ code with dual distance at least ${leftlceilfrac{d^bot}{q}rightrceil}$ using construction Y ₁. The inverse construction gives a rule for the classification of all [n, k, d] _q codes with dual distance ${d^{bot}}$ by adding ${d^bot}$ further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] _q , q =  2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.