## 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 *A*_{i} = 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 ℤ_{3} can exist.

### Over the Quaternary Field

Greenough and Hill show the nonexistence of a [32, 4, 23]-code over **F**_{4} 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 **F**_{4} 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 **F**_{4} is shown in [5, Theorem 13] by Boukliev, Daskalov and Kapralov.

### Over ℤ_{5}

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.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CBoundLPConstraints.html