Abstract An affine vector space partition of $${{\,\textrm{AG}\,}}(n,q)$$ 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.

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

Abstract 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 parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues $$k> \theta _1> \theta _2 > \theta _3$$ k > θ 1 > θ 2 > θ 3 , and that the triple $$(\theta _1, \theta _2, \theta _3)$$ ( θ 1 , θ 2 , θ 3 ) satisfies a certain homogeneous polynomial equation of degree $$s - 2$$ s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $$s = 5$$ s = 5 and $$s = 7$$ s = 7 , there are only the obvious solutions, and we conjecture this to remain true for all (odd) $$s \ge 9$$ s ≥ 9 .

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 codewords of weight $4$ have been classified, which is the case $q=2$ and $\Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $\Delta$-divisible codes of length $4\Delta$ in the projective case.

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, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of$q$. In particular, we correct a flawed proof for the original cylinder conjecture for$q = 5$and present the first proof for$q = 7$.

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.

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 system. We introduce a geometric approach to address this problem. One of the most significant advantages of this approach over the existing ones is that it allows one to derive bounds on the service rate of a code without explicitly knowing the list of all possible recovery sets. To illustrate the power of our geometric approach, we derive upper bounds on the service rates of the first order Reed-Muller codes and the simplex codes. Then, we show how these upper bounds can be achieved. Furthermore, utilizing the proposed geometric technique, we show that given the service rate region of a code, a lower bound on the minimum distance of the code can be obtained.

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 designing a coded distributed storage system storing k files where a desired service rate region $\mathcal{R}$ of the system is given and the goal is 1) to determine the minimum number of storage nodes $n(\mathcal{R})$ for serving all demand vectors inside the set $\mathcal{R}$ and 2) to design the most storage-efficient redundancy scheme with the service rate region covering the set $\mathcal{R}$. Towards this goal, we propose three general lower bounds for $n(\mathcal{R})$. Also, for k = 2, we characterize $n(\mathcal{R})$, i.e., we show that the proposed lower bounds are tight, via designing a novel storage-efficient redundancy scheme with $n(\mathcal{R})$ storage nodes and service rate region covering $\mathcal{R}$.

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.

In this article, the effective lengths of all qr-divisible linear codes over Fqwith 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 of n is non-negative. Furthermore, the maximum weight of a qr-divisible code of effective length n is at most σqr, where σ denotes the cross-sum of the Sq(r)-adic expansion of n. This result has applications in Galois geometries. A recent theorem of Nästase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension 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 Pq(n), i.e., the set of subspaces of Fqn, 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 maximum-rank distance bound for an infinite family of parameters.

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:{0,1}n→{0,1}. However, its naive encoding needs 2nbits. 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 requiringand provided a construction guaranteeing. The magnitude of the worst-case situation was 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≥2n−o(n), i.e., there is no gain from a representation complexity point of view.

Important decisions are likely made by groups of agents. Thus group decision making is very common in practice. Very transparent group aggregating rules are given by weighted voting, where each agent is assigned a weight. Here a proposal is accepted if the sum of the weights of the supporting agents meets or exceeds a given quota. We study a more general class of binary voting systems -- complete simple games -- and propose an algorithm to determine which sub classes, parameterized by the agent's type composition, are weighted.

We propose the new Top-Dog-Index to quantify the his- toric deviation of the supply data of many small branches for a commodity group from sales data. On the one hand, the common parametric assumptions on the customer de- mand distribution in the literature could not at all be sup- ported in our real-world data set. On the other hand, a reasonably-looking non-parametric approach to estimate the demand distribution for the different branches directly from the sales distribution could only provide us with statis- tically weak and unreliable estimates for the future demand. Based on real-world sales data from our industry partner we provide evidence that our Top-Dog-Index is statistically robust. Using the Top-Dog-Index, we propose a heuristics to improve the branch-dependent proportion between sup- ply and demand. Our approach cannot estimate the branch- dependent demand directly. It can, however, classify the branches into a given number of clusters according to an historic oversupply or undersupply. This classification of branches can iteratively be used to adapt the branch distri- bution of supply and demand in the future.