Integer Linear Programming Models for Subspace Codes and Finite Geomery
子空间代码和有限几何的整数线性规划模型
基本信息
- 批准号:266952998
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:2015
- 资助国家:德国
- 起止时间:2014-12-31 至 2017-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Error-correcting codes are employed in almost every form of information transmission and storage. Well known examples include communication with space probes, Digital TV (DVB), ADSL, distributed storage systems (Windows Azure) and CD player. Numerous applications in communication networks such as the Internet, mobile devices like smartphones and cloud computing, require a mathematical theory that is independent of the exact network topology. A few years ago it was recognized that the throughput rates in certain situations can significantly be improved by using multiplexed data packets, i.e., linear combinations over a suitable finite field. If also errors shall be corrected in the communication model, the set of all subspaces of a finite vector space is well suited as an alphabet. With this a subspace code is a subset of convenient subspaces. One major research direction is the question of existence and construction of good or even optimal subspace codes. This problem is up-to-the-minute and indeed the topic of the EU COST Action IC1104. An exhaustive search of subspace codes will be performed. Problems, caused by the huge number of subspaces of a finite vector space and the size of the automorphism group, have to be tackled. The construction methods previously applied in the Bayreuth research group using prescribing a subgroup of the automorphism group, will be used intensively in order to reduce the search space to a convenient size. The underlying idea is to formulate the problem as an integer linear system of equations. This description fits well with the conditions of automorphisms, since it massively reduces both, the number of variables and the number of constraints. There is reason to hope that subspace codes with good properties have some automorphisms. As an innovation, sharper integer formulations shall be found. The basic idea here is to use results and substructures of finite geometry on vector spaces. From an algorithmic point of view, some of the arguments used there correspond to adding feasible inequalities or Gomory-Chvátal cutting planes. The main idea here is to model and enumerate geometric structures, such as holes, cuts, or regulus, and the use of methods of integer linear optimization. The procedure described in "T. Honold, M. Kiermaier, and S. Kurz. Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4. 2014. to appear in the Proceedings of Fq 11" for the classification of so-called (6,3,4)_2 constant-dimension codes can be seen as an example for the new strategic approach (see appendix). Since there are very few moderate parameters in the area of subspace codes, it is justified to make a considerable effort for each of them. Nevertheless, all smaller cases shall be examined systematically. Even the improvement of barriers in some special cases can be a progress.
纠错码几乎应用于所有形式的信息传输和存储。众所周知的例子包括与空间探测器、数字电视(DVB)、ADSL、分布式存储系统(Windows Azure)和CD播放器的通信。在诸如互联网的通信网络、诸如智能手机的移动的设备和云计算中的许多应用需要独立于确切网络拓扑的数学理论。几年前,人们认识到,在某些情况下,通过使用多路复用的数据分组,即,在适当的有限域上的线性组合。如果在通信模型中也要纠正错误,则有限向量空间的所有子空间的集合非常适合作为字母表。这样,子空间码是方便子空间的子集。一个主要的研究方向是好的甚至是最优的子空间码的存在性和构造问题。这个问题是最新的,实际上是欧盟成本行动IC1104的主题。将执行子空间代码的穷举搜索。由于有限向量空间的子空间数量巨大以及自同构群的大小而引起的问题必须解决。以前应用在拜罗伊特研究小组使用规定的自同构群的子群的构造方法,将被集中使用,以减少搜索空间到一个方便的大小。其基本思想是将问题表述为整数线性方程组。这种描述非常符合自同构的条件,因为它大大减少了变量的数量和约束的数量。有理由希望具有良好性质的子空间码具有一些自同构。作为一个创新,我们将找到更清晰的整数公式,这里的基本思想是利用向量空间上有限几何的结果和子结构。从算法的角度来看,这里使用的一些参数对应于添加可行的不等式或Gomory-Chvátal切割平面。这里的主要思想是建模和枚举几何结构,如孔,切口或regulus,以及整数线性优化方法的使用。在"T. Honold,M. Kiermaier和S.库尔兹长度为6、常维数为3、最小距离为4的最优二元子空间码。2014.出现在Fq 11的会议记录中",用于对所谓的(6,3,4)_2等维编码进行分类,可以看作是新战略方法的一个例子(见附录)。由于子空间码领域的适度参数很少,因此有理由为每个参数付出相当大的努力。然而,所有较小的案件都应系统地加以审查。在某些特殊情况下,甚至障碍的改善也可以是一种进步。
项目成果
期刊论文数量(8)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
The order of the automorphism group of a binary $${\varvec{q}}$$q-analog of the Fano plane is at most two
- DOI:10.1007/s10623-017-0360-6
- 发表时间:2016-05
- 期刊:
- 影响因子:0
- 作者:Michael Kiermaier;Sascha Kurz;A. Wassermann
- 通讯作者:Michael Kiermaier;Sascha Kurz;A. Wassermann
Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6
对长度为 8、常量维度为 4、最小距离为 6 的最佳二进制子空间代码进行分类
- DOI:10.1007/s10623-018-0544-8
- 发表时间:2017-11
- 期刊:
- 影响因子:0
- 作者:D. Heinlein;T. Honold;M. Kiermaier;S. Kurz;A. Wassermann
- 通讯作者:A. Wassermann
Partial spreads and vector space partitions
部分扩散和向量空间划分
- DOI:10.1007/978-3-319-70293-3_7
- 发表时间:2016-11
- 期刊:
- 影响因子:0
- 作者:T. Honold;M. Kiermaier;S. Kurz
- 通讯作者:S. Kurz
Binary Subspace Codes in Small Ambient Spaces
- DOI:10.3934/amc.2018048
- 发表时间:2018-04
- 期刊:
- 影响因子:0
- 作者:Daniel Heinlein;Sascha Kurz
- 通讯作者:Daniel Heinlein;Sascha Kurz
Classification of large partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$PG(6,2) and related combinatorial objects
- DOI:10.1007/s00022-018-0459-6
- 发表时间:2016-06
- 期刊:
- 影响因子:0.6
- 作者:T. Honold;Michael Kiermaier;Sascha Kurz
- 通讯作者:T. Honold;Michael Kiermaier;Sascha Kurz
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Privatdozent Dr. Sascha Kurz其他文献
Privatdozent Dr. Sascha Kurz的其他文献
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