The construction of 32 new codes over the Galois field of q elements was discussed. The use of a computer program to prove the nonexistence of 51 codes in the Galois field was also discussed. The linear program for each of the 51 codes was solved using simplex method to prove the nonexistence. It was also shown that that for each nonexistence code the MacWilliams identities do not have a solution in nonnegative integers.
Daskalov, R., Hristov, P., & Metodieva, E. (2004). New minimum distance bounds for linear codes over GF(5). Discrete Mathematics, 275(1–3), 97–110. https://doi.org/10.1016/S0012-365X(03)00126-2