# cdc7-4-3-2

### binaryset_packing

Submitter Variables Constraints Density Status Group Objective MPS File
Sascha Kurz 11811 14478 1.51955e-03 open -- -289.0* 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.

# Instance Statistics

Detailed explanation of the following tables can be found here.

Size Related Properties
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
Constraint Classification Properties
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

# Structure

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

value min median mean max
Components 0.30103
Constraint % 100 100 100 100
Variable % 100 100 100 100
Score 0.00000

# Best Known Solution(s)

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

ID Objective Exact Int. Viol Cons. Viol Obj. Viol Submitter Date Description
4 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
5 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.
2 5 -285 -285 0 0 0 Edward Rothberg 2020-04-22 Obtained with Gurobi 9.0 using the solution improvement heuristic
6 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
1 3 -279 0 0 0 Edward Rothberg 2019-12-13 Obtained with Gurobi 9.0
7 2 -275 0 0 0 Robert Ashford and Alkis Vazacopoulus 2019-12-18 Found using ODH|CPlex
3 1 -260 -260 0 0 0 - 2018-10-13 Solution found during MIPLIB2017 problem selection.

# Similar instances in collection

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 20078700 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 -196.0* binary variable_bound
z26 open 17937 17937 0 0 850513 1715610 Daniel Bienstock -- -1187.0* binary variable_bound set_packing

## Reference

@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}
}

Last Update Nov 16, 2020 by Philipp Wellner
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