Sascha Kurz

University of Bayreuth · Institute of Mathematics

After the optimal parameters of additive quaternary codes of dimension $k\le 3$ have been determined there is some recent activity to settle the next case of dimension $k=3.5$. Here we complete dimension $k=3.5$ and give partial results for dimension $k=4$.

We consider the problem of error correction in a network where the errors can occur only on a proper subset of the network edges. For a generalization of the so-called Diamond Network we consider lower and upper bounds for the network’s (1-shot) capacity for fixed alphabet sizes.

A linear code C over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q$$\end{document} is called Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \us...

It has been known since the 1970’s that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of...

It has been known since the 1970's that the difference of the non-zero weights of a projective Fq-linear two-weight has to be a power of the characteristic of the underlying field. Here we study non-projective two-weight codes and e.g. show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feas...

A subcube partition is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal...

In this paper, we give a geometric construction of the three strong non-lifted (3mod5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3\mod 5)$$\end{document}-arcs in...

An affine vector space partition of AG(n,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{AG}\,}}(n,q)$$\end{document} is a set of proper affine subspaces...

We classify all q -ary Δ-divisible linear codes which are spanned by codewords of weight Δ. The basic building blocks are the simplex codes, and for q = 2 additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight...

We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $q^r$-divisible codes over $\mathbb{F}_q$. We also give a few computational results for field sizes $q>2$. Non-existence results of divisible codes with restricted column...

A vector space partition P of the projective space PG(v-1,q) is a set of subspaces in PG(v-1,q) which partitions the set of points. We say that a vector space partition P has type (v-1)^{m_{v-1}} ... 2^{m_2}1^{m_1} if precisely m_i of its elements have dimension i, where 1 <= i <= v-1. Here we determine all possible types of vector space partitions...

The Public Good index is a power index for simple games introduced by Holler and later axiomatized by Holler and Packel so that some authors also speak of the Holler–Packel index. A generalization to the class of games with transferable utility was given by Holler and Li. Here, we generalize the underlying ideas to games with several levels of appr...

We exhibit the hidden beauty of weighted voting and voting power by applying a generalization of the Penrose-Banzhaf index to social choice rules. Three players who have multiple votes in a committee decide between three options by plurality rule, Borda’s rule, and antiplurality rule, or one of the many scoring rules in between. A priori influence...

A vector space partition $\mathcal{P}$ of the projective space $\operatorname{PG}(v-1,q)$ is a set of subspaces in $\operatorname{PG}(v-1,q)$ which partitions the set of points. We say that a vector space partition $\mathcal{P}$ has type $(v-1)^{m_{v-1}} \dots 2^{m_2}1^{m_1}$ if precisely $m_i$ of its elements have dimension $i$, where $1\le i\le v...

A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions:...

In this paper, we give a geometric construction of the three strong non-lifted $(3\mod{5})$-arcs in $\operatorname{PG}(3,5)$ of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong $(t\mod{q})$-arcs in $\operatorname{PG}(r,q)$ with $t=(q+1)/2$ for all $r\ge3$ and all odd prime powers $q$.

We consider the problem of error correction in a network where the errors can occur only on a proper subset of the network edges. For a generalization of the so-called Diamond Network we consider lower and upper bounds for the network's ($1$-shot) capacity.

An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible paramete...

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over Zpm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{do...

Minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a graph.

A basic problem for constant dimension codes is to determine the maximum possible size \begin{document}$ A_q(n,d;k) $\end{document} of a set of \begin{document}$ k $\end{document}-dimensional subspaces in \begin{document}$ \mathbb{F}_q^n $\end{document}, called codewords, such that the subspace distance satisfies \begin{document}$ d_S(U,W): = 2k-2\...

How many squares are spanned by $n$ points in the plane? Here we study the corresponding maximum possible number $S_{\square}(n)$ of squares and determine the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds for $S_{\square}(n)$. Besides that a few preliminary structural results are obtained. For the related problem of the...

Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes.

A linear code over $\mathbb{F}_q$ with the Hamming metric is called $\Delta$-divisible if the weights of all codewords are divisible by $\Delta$. They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible len...

We exhibit the hidden beauty of weighted voting and voting power by applying a generalization of the Penrose-Banzhaf index to social choice rules. Three players who have multiple votes in a committee decide between three options by plurality rule, Borda's rule, antiplurality rule, or one of the scoring rules in between. A priori influence on outcom...

We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over Fq and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prov...

Binary “yes”–“no” decisions in a legislative committee or a shareholder meeting are commonly modeled as a weighted game. However, there are noteworthy exceptions. E.g., the voting rules of the European Council according to the Treaty of Lisbon use a more complicated construction. Here we want to study the question if we lose much from a practical p...

The application of flags to network coding has been introduced recently, see e.g. Liebhold et al. (Des Codes Cryptogr, 86(2):269-284, 2018). It is a variant to random linear network coding and explicit routing solutions for given networks. Here we study lower and upper bounds for the maximum possible cardinality of a corresponding flag code with gi...

We consider integers whose squares have just three decimal digits. Examples are e.g. given by $2108436491907081488939581538^2 = 4445504440405440505004450045555054500055550554550445444$ and $10100000000010401000000000101^2 = 102010000000210100200000110221001000002101002000000010201$. The aim of this paper is to summarize the current knowledge on squ...

Many real-world voting systems consist of voters that occur in just two different types. Indeed, each voting system with a {\lq\lq}House{\rq\rq} and a {\lq\lq}Senat{\rq\rq} is of that type. Here we present structural characterizations and explicit enumeration formulas for these so-called bipartite simple games.

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained.

A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2\dim(U\cap W)\ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g...

Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension k and length k+2 and f...

The proof of the non-existence of Griesmer $[104, 4, 82]_5$-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of $(t\operatorname{mod} q)$-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classifi...

The Public Good index is a power index for simple games introduced by Holler and later axiomatized by Holler and Packel, so that some authors also speak of the Holler--Packel index. A generalization to the class of games with transferable utility was given by Holler and Li. Here we generalize the underlying ideas to games with several levels of app...

The Shapley–Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called (j, k) simple games. Here we present a new axiom...

In this work private information retrieval (PIR) codes are studied. In a k-PIR code, s information bits are encoded in such a way that every information bit has k mutually disjoint recovery sets. The main problem under this paradigm is to minimize the number of encoded bits given the values of s and k, where this value is denoted by P(s, k). The ma...

We consider a fashion discounter distributing its many branches with integral multiples from a set of available lot-types. For the problem of approximating the branch and size dependent demand using those lots we propose a tailored exact column generation approach assisted by fast algorithms for intrinsic subproblems, which turns out to be very eff...

The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both measures can be exponential in the number of voters. For complete simple games it was only recently shown in O’Dwyer and Slinko (2017) that the dimension can also be exponential. Here we s...

The proof of the non-existence of Griesmer \begin{document}$ [104, 4, 82]_5 $\end{document}-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of \begin{document}$ (t\mod q) $\end{document}-arcs as a general framework for extendability results for codes and...

Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~$s$. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible para...

We classify 8-divisible binary linear codes with minimum distance 24 and small length. As an application we consider the codes associated to nodal sextics with 65 ordinary double points.

Committee decisions on more than two alternatives much depend on the adopted aggregation rule, and so does the distribution of power among committee members. We quantify how different voting methods such as pairwise majority votes, plurality voting with or without a runoff, or Borda rule map asymmetric numbers of seats, shares, voting weights, etc....

One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $\mathcal{P}_q(n)$ for a given minimum distance. The determination of the exact maximum cardinality is a very tough discrete optimization problem involving a huge number of...

We classify all $q$-ary $\Delta$-divisible linear codes which are spanned by codewords of weight $\Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codeword...

We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over $\mathbb{F}_q$ and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way,...

A major concern in cloud/edge storage systems is serving a large number of users simultaneously. The service rate region is introduced recently as an important performance metric for coded distributed systems, which is defined as the set of all data access requests that can be simultaneously handled by the system. This paper studies the problem of...

Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension $k$ and length $k+2$ a...

We show that no projective 16-divisible binary linear code of length 131 exists. This implies several improved upper bounds for constantdimension codes, used in random linear network coding, and partial spreads.

Grassmannian \({{{\mathcal {G}}}}_q(n,k)\) is the set of all k-dimensional subspaces of the vector space \({\mathbb {F}}_q^n\). Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are q-analogs of codes in the Johnson scheme, i.e. constant dimension...

We show that no projective 16-divisible binary linear code of length 131 exists. This implies several improved upper bounds for constant-dimension codes, used in random linear network coding, and partial spreads.

In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality...

Binary yes-no decisions in a legislative committee or a shareholder meeting are commonly modeled as a weighted game. However, there are noteworthy exceptions. E.g., the voting rules of the European Council according to the Treaty of Lisbon use a more complicated construction. Here we want to study the question if we loose much from a practical poin...

The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games it was only recently shown that the dimension can also be exponential. Here we show that this is also the case...

Minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a graph.

The application of flags to network coding has been introduced recently by Liebhold, Nebe, and Vazquez-Castro. It is a variant to random linear network coding and explicit routing solutions for given networks. Here we study lower and upper bounds for the maximum possible cardinality of a corresponding flag code with given parameters.

Constant dimension codes are e.g. used for error correction and detection in random linear network coding, so that constructions for these codes have achieved wide attention. Here, we improve over 150 lower bounds by describing better constructions for subspace distance 4.

A simple game (N, v) is given by a set N of n players and a partition of \(2^N\) into a set \(\mathcal {L}\) of losing coalitions L with value \(v(L)=0\) that is closed under taking subsets and a set \(\mathcal {W}\) of winning coalitions W with value \(v(W)=1\). We let \(\alpha = \min _{p\geqslant {\varvec{0}}, p\ne {\varvec{0}}}\max _{W\in \mathc...

It is well known that the Penrose–Banzhaf index of a weighted game can differ starkly from corresponding weights. Limit results are quite the opposite, i.e., under certain conditions the power distribution approaches the weight distribution. Here we provide parametric examples that give necessary conditions for the existence of limit results for th...

We improve on the lower bound of the maximum number of planes in PG(8,q)≅Fq9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {PG}}(8,q)\cong \mathbb {F}_q^{9}$$\e...

We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements. We validate a correct implementation of the algorithms with known classification results from the literature,...

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called $(j,k)$ simple games. Here we present a new axio...

Service rate is an important, recently introduced, performance metric associated with distributed coded storage systems. Among other interpretations, it measures the number of users that can be simultaneously served by the storage system. We introduce a geometric approach to address this problem. One of the most significant advantages of this appro...

In this article, the effective lengths of all qr-divisible linear codes over Fq with a non-negative integer r are determined. For that purpose, the Sq(r)-adic expansion of an integer n is introduced. It is shown that there exists a qr-divisible Fq-linear code of effective length n if and only if the leading coefficient of the Sq(r)-adic expansion o...

In this work private information retrieval (PIR) codes are studied. In a $k$-PIR code, $s$ information bits are encoded in such a way that every information bit has $k$ mutually disjoint recovery sets. The main problem under this paradigm is to minimize the number of encoded bits given the values of $s$ and $k$, where this value is denoted by $P(s,...

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called $(j,k)$~simple games. Here we present a new axio...

A committee's decisions on more than two alternatives much depend on the adopted voting method, and so does the distribution of power among the committee members. We investigate how different aggregation methods such as plurality runoff, Borda count, or Copeland rule map asymmetric numbers of seats, shares, voting weights, etc. to influence on outc...

We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.

A projective linear code over $\mathbb{F}_q$ is called $\Delta$-divisible if all weights of its codewords are divisible by $\Delta$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $\mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial sp...

We present an algorithm for the classification of linear codes over finite fields, based on lattice point enumeration. We validate a correct implementation of our algorithm with known classification results from the literature, which we partially extend to larger ranges of parameters.

We consider a fashion discounter distributing its many branches with integral multiples from a set of available lot-types. For the problem of approximating the branch and size dependent demand using those lots we propose a tailored exact column generation approach assisted by fast algorithms for intrinsic subproblems, which turns out to be very eff...

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$ a prime, and more generally over chain rings of depth $m$, and with a residue field of size $q$, a prime power. Infinite families of examples are built from Kerdock and generalized Teichm\"uller co...

In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size $A_q(n, d; k)$ of codes whose codewords are $k$-subspaces of $\mathbb{F}_q^n$ with minimum subspace distance $d$. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow...

Many binary collective choice situations can be described as weighted simple voting games. We introduce weighted committee games to model decisions on an arbitrary number of alternatives in analogous fashion. We compare the effect of different voting weights (shareholdings, party seats, etc.) under plurality, Borda, Copeland, and antiplurality rule...

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45–51, and ≥60.

The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. K\"{o}tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension...

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

A well-known class of objects in combinatorial design theory are group divisible designs. Here, we introduce the q-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, q-Steiner systems, packing designs and qr-divisible projective sets. We give necessary conditions for the existence of q-an...

We improve on the lower bound of the maximum number of planes in $\operatorname{PG}(8,q)\cong\F_q^{9}$ pairwise intersecting in at most a point. In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+ 2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1$. This result is obtained via a more general construction strategy, which also yields other impro...

The Shapley-Shubik index was designed to evaluate the power distribution in committee systems drawing binary decisions and is one of the most established power indices. It was generalized to decisions with more than two levels of approval in the input and output. In the limit we have a continuum of options. For these games with interval decisions w...

Constant dimension codes are used for error control in random linear network coding, so that constructions for these codes with large cardinality have achieved wide attention in the last decade. Here, we improve the so-called linkage construction and obtain several parametric series of improvements for code the sizes.

The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length $n=46$ with known bounds $19\le d\le 20$. Here we present a $[46,9,20]_2$ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Ad...

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric (quota) games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is kno...

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In Finite Geometry terms the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a...

We discuss possible criteria that may qualify or disqualify power indices for applications. Instead of providing final answers we merely ask questions that are relevant from our point of view and summarize some material from the literature.