Submitter | Variables | Constraints | Density | Status | Group | Objective | MPS File |
---|---|---|---|---|---|---|---|

Sascha Kurz | 11811 | 14478 | 1.51955e-03 | open | – | -296.0* | cdc7-4-3-2.mps.gz |

Codes for Networkcoding A constant dimension code with parameters n, k, d and q is a collection of k-dimensional subspaces of the n-dimensional vector space \(GF(q)^n\) over a finite field with q elements, called codewords, such that the dimension of the intersection of each pair of different k-dimensional subspaces is at most \(k-d/2\). Let \(A_q(n,d;k)\) denote the maximum number of codewords. For instance cdc6-4-3-2 \(A_2(6,4;3)=77\) is known , while \(333 \le A_2(7,4;3) \le 381\) for instance cdc7-4-3-2 are the tightest known bounds, see e.g. . A code of size 381 would correspond to a putative binary q-analog of the Fano plane (finite projective plane of order 2 with 7 points and lines). More bounds are available at http://subspacecodes.uni-bayreuth.de.

Detailed explanation of the following tables can be found here.

Original | Presolved | |
---|---|---|

Variables | 11811 | 11811 |

Constraints | 14478 | 14478 |

Binaries | 11811 | 11811 |

Integers | 0 | 0 |

Continuous | 0 | 0 |

Implicit Integers | 0 | 0 |

Fixed Variables | 0 | 0 |

Nonzero Density | 0.00151955 | 0.00151955 |

Nonzeroes | 259842 | 259842 |

Original | Presolved | |
---|---|---|

Total | 14478 | 14478 |

Empty | 0 | 0 |

Free | 0 | 0 |

Singleton | 0 | 0 |

Aggregations | 0 | 0 |

Precedence | 0 | 0 |

Variable Bound | 0 | 0 |

Set Partitioning | 0 | 0 |

Set Packing | 14478 | 14478 |

Set Covering | 0 | 0 |

Cardinality | 0 | 0 |

Invariant Knapsack | 0 | 0 |

Equation Knapsack | 0 | 0 |

Bin Packing | 0 | 0 |

Knapsack | 0 | 0 |

Integer Knapsack | 0 | 0 |

Mixed Binary | 0 | 0 |

General Linear | 0 | 0 |

Indicator | 0 | 0 |

Available nonzero structure and decomposition information. Further information can be found here.

Decomposed structure of original problem (dec-file)

Decomposed structure after trivial presolving (dec-file)

value | min | median | mean | max | |
---|---|---|---|---|---|

Components | 0.30103 | ||||

Constraint % | 100 | 100 | 100 | 100 | |

Variable % | 100 | 100 | 100 | 100 | |

Score | 0.00000 |

Find solutions below. Download the archive containing all solutions from the Download page.

`## Warning in lapply(df["exactobjval"], as.numeric): NAs introduced by coercion`

ID | Objective | Exact | Int. Viol | Cons. Viol | Obj. Viol | Submitter | Date | Description |
---|---|---|---|---|---|---|---|---|

9 | -296 | -296 | 0 | 0 | 0 | Taishi Maeda and Ryuhei Miyashiro | 2023-02-06 | Obtained with Fixstars Amplify Annealing Engine, a GPU-based QUBO solver. |

8 | -294 | -294 | 0 | 0 | 0 | Peng Lin, Shaowei Cai, Mengchuan Zou, Jinkun Lin | 2023-05-03 | Computed by local search based on the paper Peng Lin, Shaowei Cai, Mengchuan Zou, and Jinkun Lin. “New Characterizations and Efficient Local Search for General Integer Linear Programming”, arXiv preprint arXiv:2305.00188 (2023). |

7 | -289 | -289 | 0 | 0 | 0 | Yuji Koguma | 2020-06-24 | Obtained with a Tabu Search algorithm based on the paper, Koji Nonobe, Toshihide Ibaraki: "An Improved Tabu Search Method For The Weighted Constraint Satisfaction Problem," INFOR, Vol.39, No.2, pp.131-151, 2001 |

6 | -288 | -288 | 0 | 0 | 0 | Shunsuke Kamiya | 2020-05-17 | Computed with weighting local search with exploiting variable associations (WLS) ([1]) [1] Umetani, Shunji. “Exploiting variable associations to configure efficient local search algorithms in large-scale binary integer programs.” European Journal of Operational Research 263.1 (2017): 72-81. |

5 | -285 | -285 | 0 | 0 | 0 | Edward Rothberg | 2020-04-22 | Obtained with Gurobi 9.0 using the solution improvement heuristic |

4 | -282 | -282 | 0 | 0 | 0 | Frederic Didier | 2020-01-22 | Obtained with Google OR-tools using 8 Threads through generating subproblems by fixing part of the current solution and trying to solve them with a sub CP-SAT solver |

3 | -279 | 0 | 0 | 0 | Edward Rothberg | 2019-12-13 | Obtained with Gurobi 9.0 | |

2 | -275 | 0 | 0 | 0 | Robert Ashford and Alkis Vazacopoulus | 2019-12-18 | Found using ODH|CPlex | |

1 | -260 | -260 | 0 | 0 | 0 | - | 2018-10-13 | Solution found during MIPLIB2017 problem selection. |

The following instances are most similar to cdc7-4-3-2 in the collection. This similarity analysis is based on 100 scaled instance features describing properties of the variables, objective function, bounds, constraints, and right hand sides.

Instance | Status | Variables | Binaries | Integers | Continuous | Constraints | Nonz. | Submitter | Group | Objective | Tags |
---|---|---|---|---|---|---|---|---|---|---|---|

cod105 | easy | 1024 | 1024 | 0 | 0 | 1024 | 57344 | MIPLIB submission pool | – | -12 | benchmark binary benchmark_suitable set_packing |

a2864-99blp | open | 200787 | 200787 | 0 | 0 | 22117 | 20078717 | Daniel Heinlein | selofsubspaces | -257* | binary set_packing invariant_knapsack |

sorrell8 | easy | 2046 | 2046 | 0 | 0 | 18944 | 37888 | Toni Sorrell | independentset | -350 | binary decomposition variable_bound |

sorrell7 | open | 2048 | 2048 | 0 | 0 | 78848 | 157696 | Toni Sorrell | independentset | -198.0* | binary variable_bound |

z26 | open | 17937 | 17937 | 0 | 0 | 850513 | 1715613 | Daniel Bienstock | – | -1195.0* | binary variable_bound set_packing |

```
@article{honold2015optimal,
title={Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4},
author={Honold, Thomas and Kiermaier, Michael and Kurz, Sascha},
journal={Contemporary Mathematics},
volume={632},
pages={157--172},
year={2015},
}
@incollection{kohnert2008construction,
title={Construction of large constant dimension codes with a prescribed minimum distance},
author={Kohnert, Axel and Kurz, Sascha},
booktitle={Mathematical Methods in Computer Science},
volume={5393},
pages={31--42},
year={2008},
publisher={Springer}
}
```

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