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Sascha Kurz

Publications and source records attributed to Sascha Kurz.

At least 109 records · Page 6Linked to original sources
arXiv2017

Coset Construction for Subspace Codes

One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in $\operatorname{PG}(n,q)$, i.e., the set of subspaces of $\mathbb{F}_q^n$, for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the MRD bound for an infinite family of parameters.

math.CO↗
arXiv2017

Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

math.CO↗
arXiv2017

Upper bounds for partial spreads

A partial $t$-spread in $\mathbb{F}_q^n$ is a collection of $t$-dimensional subspaces with trivial intersection such that each non-zero vector is covered at most once. We present some improved upper bounds on the maximum sizes.

math.CO↗
arXiv2017

A new upper bound for subspace codes

It is shown that the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=4$, and constant dimension $k=4$ is at most $272$. In Finite Geometry terms, the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a point, is at most $272$. Previously, the best known upper bound $A_2(8,6;4)\le 289$ was implied by the Johnson bound and the maximum size $A_2(7,6;3)=17$ of partial plane spreads in $\operatorname{PG}(6,2)$. The result was obtained by combining the classification of subspace codes with parameters $(7,17,6;3)_2$ and $(7,34,5;\{3,4\})_2$ with integer linear programming techniques. The classification of $(7,33,5;\{3,4\})_2$ subspace codes is obtained as a byproduct.

math.CO↗
arXiv2017

Projective divisible binary codes

For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possible minimum distance, since the set of holes of a partial spread of $r$-flats in $\operatorname{PG}(v-1,\mathbb{F}_q)$ corresponds to a $q^r$-divisible code with $k\leq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.

math.CO↗
arXiv2017

The nucleolus of large majority games

Members of a shareholder meeting or legislative committee have greater or smaller voting power than meets the eye if the nucleolus of the induced majority game differs from the voting weight distribution. We establish a new sufficient condition for the weight and power distributions to be equal; and we characterize the limit behavior of the nucleolus in case all relative weights become small.

math.OC↗
arXiv2017

Partial spreads and vector space partitions

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.

math.CO↗
arXiv2017

Characterization of threshold functions: state of the art, some new contributions and open problems

This paper has a twofold scope. The first one is to clarify and put in evidence the isomorphic character of two theories developed in quite different fields: on one side, threshold logic, on the other side, simple games. One of the main purposes in both theories is to determine when a simple game is representable as a weighted game, which allows a very compact and easily comprehensible representation. Deep results were found in threshold logic in the sixties and seventies for this problem. However, game theory has taken the lead and some new results have been obtained for the problem in the last two decades. The second and main goal of this paper is to provide some new results on this problem and propose several open questions and conjectures for future research. The results we obtain depend on two significant parameters of the game: the number of types of equivalent players and the number of types of shift-minimal winning coalitions.

cs.GT↗
arXiv2016

Fair Representation and a Linear Shapley Rule

When delegations to an assembly or council represent differently sized constituencies, they are often allocated voting weights which increase in population numbers (EU Council, US Electoral College, etc.). The Penrose square root rule (PSRR) is the main benchmark for fair representation of all bottom-tier voters in the top-tier decision making body, but rests on the restrictive assumption of independent binary decisions. We consider intervals of alternatives with single-peaked preferences instead, and presume positive correlation of local voters. This calls for a replacement of the PSRR by a linear Shapley rule: representation is fair if the Shapley value of the delegates is proportional to their constituency sizes.

cs.GT↗
arXiv2016

On the Construction of High Dimensional Simple Games

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function $χ\colon\{0,1\}^n\rightarrow \{0,1\}$. However, its naive encoding needs $2^n$ bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using $n$ weights and one threshold. For heterogeneous agents, one can represent $χ$ as an intersection of $k$ threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring $k\ge 2^{\frac{n}{2}-1}$ and provided a construction guaranteeing $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-o(n)}$. The magnitude of the worst-case situation was thought to be determined by Elkind et al.~in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number $k$ for a subclass of voting systems. As an application, we give a construction for $k\ge 2^{n-o(n)}$, i.e., there is no gain from a representation complexity point of view.

cs.GT↗
arXiv2016

Improved upper bounds for partial spreads

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ was known for the cases $n\equiv 0\pmod k$, $n\equiv 1\pmod k$ and $n\equiv 2\pmod k$ with the additional requirements $q=2$ and $k=3$. We completely resolve the case $n\equiv 2\pmod k$ for the binary case $q=2$.

math.CO↗
arXiv2016

Computing the Power Distribution in the IMF

The International Monetary Fund is one of the largest international organizations using a weighted voting system. The weights of its 188 members are determined by a fixed amount of basic votes plus some extra votes for so-called Special Drawing Rights (SDR). On January 26, 2016, the conditions for the SDRs were increased at the 14th General Quota Review, which drastically changed the corresponding voting weights. However, since the share of voting weights in general is not equal to the influence, of a committee member on the committees overall decision, so-called power indices were introduced. So far the power distribution of the IMF was only computed by either approximation procedures or smaller games than then entire Board of Governors consisting of 188 members. We improve existing algorithms, based on dynamic programming, for the computation of power indices and provide the exact results for the IMF Board of Governors before and after the increase of voting weights. Tuned low-level details of the algorithms allow the repeated routine with sparse computational resources and can of course be applied to other large voting bodies. It turned out that the Banzhaf power shares are rather sensitive to changes of the quota.

cs.GT↗
arXiv2016

Dimension and codimension of simple games

This paper studies the complexity of computing a representation of a simple game as the intersection (union) of weighted majority games, as well as, the dimension or the codimension. We also present some examples with linear dimension and exponential codimension with respect to the number of players.

cs.GT↗
arXiv2015

The Maximum Scatter TSP on a Regular Grid

In the maximum scatter traveling salesman problem the objective is to find a tour that maximizes the shortest distance between any two consecutive nodes. This model can be applied to manufacturing processes, particularly laser melting processes. We extend an algorithm by Arkin et al. that yields optimal solutions for nodes on a line to a regular $m \times n$-grid. The new algorithm $\textsc{Weave}(m,n)$ takes linear time to compute an optimal tour in some cases. It is asymptotically optimal and a $\frac{\sqrt{10}}{5}$-approximation for the $3\times 4$-grid, which is the worst case.

cs.DM↗
arXiv2015

Key drivers of EU budget allocation: Does power matter?

We examine the determinants of the EU budget expenditures allocation among different countries. In line with earlier literature, we consider two alternative explanations for the EU budget distribution: political power vs. needs view. Extending the original data set from Kauppi and Widgrén (2004), we analyze the robustness of their predictions when applying a different measure of power and more sophisticated econometric techniques. We conclude that the nucleolus is a good alternative to the Shapley-Shubik index in distributive situations such as the case of EU budget allocation. Our results also show that when explaining budget shares, the relative weight of political power based on the nucleolus is lower than the predictions of previous studies based on the Shapley-Shubik index.

econ.GN↗
arXiv2015

Representation-Compatible Power Indices

This paper studies power indices based on average representations of a weighted game. If restricted to account for the lack of power of dummy voters, average representations become coherent measures of voting power, with power distributions being proportional to the distribution of weights in the average representation. This makes these indices representation-compatible, a property not fulfilled by classical power indices. Average representations can be tailored to reveal the equivalence classes of voters defined by the Isbell desirability relation, which leads to a pair of new power indices that ascribes equal power to all members of an equivalence class.

cs.GT↗