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We present various new constructions and bounds for arcs in projective Hjelmslev planes over finite chain rings of nilpotency index 2. For the chain rings of cardinality at most 25 we give updated tables with the best known upper and lower bounds for the maximum size of such arcs., Comment: 53 pages, 6 tables, 5 figures

In 1982, Cameron and Liebler investigated certain "special sets of lines" in PG(3,q), and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these "Cameron-Liebler line classes" got a lot of attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the "Johnson" and "q-Johnson schemes". In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of "designs" and "antidesigns" in association schemes, which is connected to Delsarte's concept of "design-orthogonality". This leads to a natural definition of the "degree" and the "weights" of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree t will be proven.

Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their $q$-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the $m$-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for $t > 1$ were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher's inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength $t=2$, resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over $\mathbb{F}_2$ and $\mathbb{F}_3$.

In 1985, Janko and Tran Van Trung published an algorithm for constructing symmetric designs with prescribed automorphisms. This algorithm is based on the equations by Dembowski (1958) for tactical decompositions of point-block incidence matrices. In the sequel, the algorithm has been generalized and improved in many articles. In parallel, higher incidence matrices have been introduced by Wilson in 1982. They have proven useful for obtaining several restrictions on the existence of designs. For example, a short proof of the generalized Fisher's inequality makes use of these incidence matrices. In this paper, we introduce a unified approach to tactical decompositions and incidence matrices. It works for both combinatorial and subspace designs alike. As a result, we obtain a generalized Fisher's inequality for tactical decompositions of combinatorial and subspace designs. Moreover, our approach is explored for the construction of combinatorial and subspace designs of arbitrary strength.

The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this module for all prime power values of $q$. The shortest $\text{PGL}(2,q^3)$-orbit $O$ gives the Desarguesian ovoid in $Q^+(7,q)$ for even $q$ and it is known to give a complete partial ovoid of the symplectic polar space $W(7,q)$ for odd~$q$. We determine the hyperplane sections of $O$. As a corollary, we obtain the parameters $[q^3+1,8,q^3-q^2-q]_q$ and the weight distribution of the associated $\mathbb{F}_q$-linear code $C_O$ and the parameters $[q^3+1,q^3-7,5]_q$ of the dual code $C_O^\perp$ for $q \ge 4$. We also show that both codes $C_O$ and $C_O^\perp$ are length-optimal for all prime power values of $q$.

We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We extend this result classifying the irreducible components of all nodal K3 surfaces in the same way, and we fully classify their extended codes. In this classification there are some sporadic cases, obtain through projection from a node. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a `Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph., Comment: 210 pages. v2 minor changes to text, added ancillary files containing supporting computer code. The new version contains some correction, completes some linear algebra proofs of code classification, adds a table showing the Kummer genealogy tree of nodal quartic surfaces. Adds also new references

Arguably, the most important open problem in the theory of $q$-analogs of designs is the question for the existence of a $q$-analog $D$ of the Fano plane. It is undecided for every single prime power value $q \geq 2$. A point $P$ is called an $\alpha$-point of $D$ if the derived design of $D$ in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must always exist at least one non-$\alpha$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$\alpha$-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $\alpha$-points implies the existence of a partiton of the symplectic generalized quadrangle $W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes $q$ and all even values of $q$.

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 $k > \theta_1 > \theta_2 > \theta_3$, and that the triple $(\theta_1, \theta_2, \theta_3)$ satisfies a certain homogeneous polynomial equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we conjecture this to remain true for all (odd) $s \ge 9$., Comment: 42 pages

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., Comment: 12 pages; typos corrected

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 spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes., Comment: 38 pages

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., Comment: 8 pages, 1 table

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 codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear., Comment: 28 pages, 6 tables

It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths $15$, $16$, $30$, $31$, $32$, $45$--$51$, and $\ge 60$., Comment: 12 pages, 4 tables

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., Comment: 16 pages, typos corrected

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, design packings and $q^r$-divisible projective sets. We give necessary conditions for the existence of $q$-analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a $(6,3,2,2)_2$ group divisible design over $\operatorname{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $\operatorname{GF}(2)^6$ is covered at most twice., Comment: 18 pages, 3 tables, typos corrected

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$., Comment: 12 pages, typos corrected

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 point, is $257$. The result was obtained by combining the classification of substructures with integer linear programming techniques. This implies that the maximum size $A_2(8,6)$ of a binary mixed-dimension code of packet length $8$ and minimum subspace distance $6$ is $257$ as well., Comment: 17 pages, 3 tables

We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., $333 \le A_2(7,4;3)$, which improves the previous best known lower bound of 329. Moreover, if a code with these parameters has at least 333 elements, its automorphism group is in one of $31$ conjugacy classes. This is achieved by a more general technique for an exhaustive search in a finite group that does not depend on the enumeration of all subgroups., Comment: 18 pages; typos corrected

In this article, the effective lengths of all $q^r$-divisible linear codes over $\mathbb{F}_q$ with a non-negative integer $r$ are determined. For that purpose, the $S_q(r)$-adic expansion of an integer $n$ is introduced. It is shown that there exists a $q^r$-divisible $\mathbb{F}_q$-linear code of effective length $n$ if and only if the leading coefficient of the $S_q(r)$-adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^r$-divisible code of effective length $n$ is at most $\sigma q^r$, where $\sigma$ denotes the cross-sum of the $S_q(r)$-adic expansion of $n$. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase 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., Comment: 17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace 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 $q=2$., Comment: 10 pages, 3 tables

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., Comment: 30 pages, 1 table

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$., Comment: 31 pages, 9 tables

It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$., Comment: 10 pages

In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$. The proof is constructive and consists of two parts. First, we give a computer construction for an $\operatorname{LS}_2[3](2,4,8)$, which is a partition of the set of all $4$-dimensional subspaces of an $8$-dimensional vector space over the binary field into three disjoint $2$-$(8, 4, 217)_2$ subspace designs. Together with the already known $\operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Gra{\ss}mannian into joins yields a construction for the claimed large sets.

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge., Comment: 44 pages, 6 tables, 7 screenshots

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called \emph{Main Problem of Subspace Coding} is to determine the maximum size $A_q(v,d)$ of a code in $\operatorname{PG}(v-1,\mathbb{F}_q)$ with minimum subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$. For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$ (all $q$), $A_2(7,5)=34$. We also provide an exposition of the known determination of $A_q(v,2)$, and a table with exact results and bounds for the numbers $A_2(v,d)$, $v\leq 7$., Comment: 35 pages, 2 tables, typo corrected

A set $\mathcal{F}_q$ of $3$-dimensional subspaces of $\mathbb{F}_q^7$, the $7$-dimensional vector space over the finite field $\mathbb{F}_q$, is said to form a $q$-analogue of the Fano plane if every $2$-dimensional subspace of $\mathbb{F}_q^7$ is contained in precisely one member of $\mathcal{F}_q$. The existence problem for such $q$-analogues remains unsolved for every single value of $q$. Here we report on an attempt to construct such $q$-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative $q$-analogue. In particular we find a ternary subspace code of new record size $6977$, and we are able to construct a binary subspace code of the largest currently known size $329$ in an entirely computer-free manner., Comment: 37 pages, results were presented in part at Alcoma15

Maximal arcs in small projective Hjelmslev geometries are classified up to isomorphism, and the parameters of the associated codes are determined.

In this paper a 2-arc of size 21 in the projective Hjelmslev plane PHG(2,\Z_25) and a 2-arc of size 22 in PHG(2,\F_5[X]/(X^2)) are given. Both arcs are bigger than the 2-arcs previously known in the respective plane. Furthermore, we will give some information on the geometrical structure of the arcs.

It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14,6,6]-code which has higher minimum distance than any linear binary [14,6]-code, can be constructed from a maximal 2-arc in the projective Hjelmslev plane over Z_4.

In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from \$\alpha\$-circulant matrices. For a non-trivial ideal I

Using integer linear programming and table-lookups we prove that there is no binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$, $[772,11,384]$, and $[836,11,416]$ is shown. Our work is motivated by the recent construction of the extended dualized Kerdock code $\hat{\mathcal{K}}^*_{6}$, which is a $\mathbb{Z}_4$-linear code having a non-linear binary Gray image with the parameters $(1988,2^{12},992)$. By our result, the code $\hat{\mathcal{K}}^*_{6}$ can be added to the small list of $\mathbb{Z}_4$-codes for which it is known that the Gray image is better than any binary linear code.

The smallest set of admissible parameters of a $q$-analog of a Steiner system is $S_2[2,3,7]$. The existence of such a Steiner system -- known as a binary $q$-analog of the Fano plane -- is still open. In this article, the automorphism group of a putative binary $q$-analog of the Fano plane is investigated by a combination of theoretical and computational methods. As a conclusion, it is either rigid or its automorphism group is cyclic of order $2$, $3$ or $4$. Up to conjugacy in $\operatorname{GL}(7,2)$, there remains a single possible group of order $2$ and $4$, respectively, and two possible groups of order $3$. For the automorphisms of order $2$, we give a more general result which is valid for any binary $q$-Steiner triple system.

Based on results in finite geometry we prove the existence of MRD codes in (F_q)_(n,n) with minimum distance n which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.