## Linear Programming Bound with Additional Constraints on the Weights

The linear programming bound for codes can be strengthened by taking into account that some weights cannot appear in a code with certain parameters. This observations yields to additional bounds of the form Ai = 0 for all iI, where I is a subset of {d + 1,…, s}.

### Over the Binary Field

The nonexistence of the following codes is established by Daskalov and Kapralov:

 Non-existent Code Reference [43, 17, 14] [1, Theorem 2] [56, 18, 20] [1, Theorem 2] [61, 19, 22] [1, Theorem 2] [56, 29, 14] [1, Theorem 2] [47, 17, 16] [1, Theorem 3]

### Over the Ternary Field

In [2] Daskalov and Metodieva use this method for showing that neither a [105, 6, 68]-code nor a [230, 6, 152]-code over 3 can exist.

### Over the Quaternary Field

Greenough and Hill show the nonexistence of a [32, 4, 23]-code over F4 in [3, Theorem 3.9], the non-existence of a [58, 4, 43]-code in [3, Theorem 3.12].

In [4] Daskalov and Metodieva use this method for showing that codes with the following parameters over F4 cannot exist: [29, 5, 20], [58, 5, 42], [62, 5, 45], [87, 5, 64], [108, 5, 80], [188, 5, 140], [192, 5, 143], [225, 5, 168], [241, 5, 180], [245, 5, 183], and [250, 5, 187]. The nonexistence of a [240, 5, 179]-code over F4 is shown in [5, Theorem 13] by Boukliev, Daskalov and Kapralov.

### Over ℤ5

The nonexistence of the following codes is established by Daskalov and Gulliver:

 Non-existent Code I Reference [75, 8, 55] {56, 61, 62, 66} [6, Theorem 5] [80, 8, 59] {61, 66, 67, 71} [6, Theorem 5]

### References

 [1] Rumen N. Daskalov and Stoyan N. Kapralov.New minimum distance bounds for certain binary linear codes.IEEE Transactions on Information Theory, 38(6):1795–1796, November 1992.doi:10.1109/18.165453 [2] Rumen N. Daskalov and Elena Metodieva.The nonexistence of ternary [105, 6, 68] and [230, 6, 152] codes.Discrete Mathematics, 286(3):225–232, September 2004.doi:10.1016/j.disc.2004.06.002 [3] P. P. Greenough and Raymond Hill.Optimal linear codes over GF(4).Discrete Mathematics, 125(1–3):187–199, February 1994.doi:10.1016/0012-365X(94)90160-0 [4] Rumen N. Daskalov and Elena Metodieva.The nonexistence of some five-dimensional quaternary linear codes.IEEE Transactions on Information Theory, 41(2):581–583, March 1995.doi:10.1109/18.370155 [5] Iliya G. Boukliev, Rumen N. Daskalov, and Stoyan N. Kapralov.Optimal quaternary linear codes of dimension five.IEEE Transactions on Information Theory, 42(4):1228–1235, July 1996.doi:10.1109/18.508846 MR1445641 (98b:94017) [6] Rumen N. Daskalov and T. Aaron Gulliver.New minimum distance bounds for linear codes over small fields.Problems of Information Transmission, 37(3):206–215, July 2001.doi:10.1023/A:1013873906597