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 i ∈ I, 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 ℤ3 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 |
Copyright
Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Linear Programming Bound with Additional Constraints on the Weights.”
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Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CBoundLPConstraints.html