Exceptions and characterization results for type‐1 λ‐designs

Let $X$ be a finite set with $v$ elements, called points and $\beta$ be a family of subsets of $X$, called blocks. A pair $\left(X,\beta \right)$ is called $\lambda$‐design whenever $\mid \beta \mid =\mid X\mid$ and

• 1. for all $B_i,B_j\in \beta ,i\ne j,\mid B_i\cap B_j\mid =\lambda$;
• 2. for all $B_j\in \beta ,\mid B_j\mid =k_j>\lambda$, and not all $k_j$ are equal.

The only known examples of $\lambda$‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$‐designs are type‐1. Let $r,r^{*}?(r>r^{*})$ be replication numbers of a $\lambda$‐design $D=\left(X,\beta \right)$ and $g=\text{gcd}\left(r?1,r^{*}?1\right),m=\text{gcd}\left(\left(r?r^{*}\right)∕g,\lambda \right)$, and $m\prime =m$, if $m$ is odd and $m\prime =m∕2$, otherwise. For distinct points $x$ and $y$ of $D$, let $\lambda \left(x,y\right)$ denote the number of blocks of $X$ containing $x$ and $y$. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if $\frac{r\left(r?1\right)}{\left(v?1\right)}?\frac{k\left(r?r^{*}\right)}{m\prime \left(v?1\right)}$ are not integers for $k=1,2,\text{…},m\prime ?1$, then $D$ is type‐1. As an application of these results, we show that for fixed positive integer $\theta$ there are finitely many nontype‐1 $\lambda$‐designs with $r=r^{*}+\theta$. If $r?r^{*}=27$ or $r?r^{*}=4p$ and $r^{*}\ne {\left(p?1\right)}^2$, or $v=7p+1$ such that $p?1,13\left(\mathrm{mod}21\right)$ and $p?4,9,19,24\left(\mathrm{mod}35\right)$, where $p$ is a positive prime, then $D$ is type‐1. We further obtain several inequalities involving $\lambda (x,y)$, where equality holds if and only if D is type‐1.     

Group divisible designs with block size four and type gum1—III

We deal with group divisible designs (GDDs) that have block size four and group type $g^u{m}^1$, where $g\equiv 2$ or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type $g^u{m}^1$ are sufficient when $g$ = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for $g=2^t{q}^s$, $q$ = 19, 23, 25, 29, 31, $s$, $t=1,2,\dots$, as well as for $g=2^{t}q$, $q$ = 2, 5, 7, 11, 13, 17, $t=1,2,\dots$, with possible exceptions $56^9{m}^1$, $80^9{m}^1$ and $112^9{m}^1$ for a few large values of $m$.     

Short k‐radius sequences, k‐difference sequences and universal cycles

An $n$‐ary $k$‐radius sequence is a finite sequence of elements taken from an alphabet of size $n$ in which any two distinct elements occur within distance $k$ of each other somewhere in the sequence. The study of constructing short $k$‐radius sequences was motivated by some problems occurring in large data transfer. Let $f_k\left(n\right)$ be the shortest length of any $n$‐ary $k$‐radius sequence. We show that the conjecture $f_k\left(n\right)=\frac{n^2}{2k}+O\left(n\right)$ by Bondy et al is true for $k\le 4$, and determine the exact values of $f_2\left(n\right)$ for new infinitely many $n$. Further, we investigate new sequences which we call $k$‐difference, as they are related to $k$‐radius sequences and seem to be interesting in themselves. Finally, we answer a question about the optimal length of packing and covering analogs of universal cycles proposed by D?bski et al.     

New lower bounds for partial k‐parallelisms

Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V\left(n,q\right)$ is an $n$‐dimensional space over the finite field ${\mathbb{F}}_q$. A $k$‐spread is a $(q^n?1)/(q^k?1)$‐set of $k$‐dimensional subspaces of $V\left(n,q\right)$ such that each nonzero vector is contained in exactly one element of it. A partial $k$‐parallelism in $V\left(n,q\right)$ is a set of pairwise disjoint $k$‐spreads. As the number of $k$‐dimensional subspaces in $V\left(n,q\right)$ is ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$, there are at most ${\left[\genfrac{}{}{0ex}{}{n?1}{k?1}\right]}_q$ spreads in a partial $k$‐parallelism. By studying the independence numbers of Cayley graphs associated to a special type of partial $k$‐parallelisms in $V\left(n,q\right)$, we obtain new lower bounds for partial $k$‐parallelisms. In particular, we show that there exist at least $\frac{q^k?1}{q^n?1}{\left[\genfrac{}{}{0ex}{}{n?1}{k?1}\right]}_q$ pairwise disjoint $k$‐spreads in $V\left(n,q\right)$.     

Nonsymmetric 2‐(v,k,λ) designs,with (r,λ) = 1, admitting a solvable flag‐transitive automorphism group of affine type

Nonsymmetric $2‐\left(v,k,\lambda \right)$ designs, with $\left(r,\lambda \right)=1$, admitting a solvable flag‐transitive automorphism group of affine type not contained in $A\mathrm{\Gamma }{L}_1\left(v\right)$ are classified.     

Reconfiguration of subspace partitions

Let $q$ be a fixed prime power and let $V\left(n,q\right)$ denote a vector space of dimension $n$ over the Galois field with $q$ elements. A subspace partition (also called “vector space partition”) of $V\left(n,q\right)$ is a collection of subspaces of $V\left(n,q\right)$ with the property that every nonzero element of $V\left(n,q\right)$ appears in exactly one of these subspaces. Given positive integers $a,b,n$ such that $1\le a<b<n$, we say a subspace partition of $V\left(n,q\right)$ has type $a^x{b}^y$ if it is composed of $x$ subspaces of dimension $a$ and $y$ subspaces of dimension $b$. Let $c=\text{gcd}\left(a,b\right)$. In this paper, we prove that if $b$ divides $n$, then one can (algebraically) construct every possible subspace partition of $V\left(n,q\right)$ of type $a^x{b}^y$ whenever $y\ge \left(q^e?1\right)∕\left(q^b?1\right)$, where $0\le e<ab∕c$ and $n\equiv e\left(\mathrm{mod}ab∕c\right)$. Our construction allows us to sequentially reconfigure batches of $\left(q^a?1\right)∕\left(q^c?1\right)$ subspaces of dimension $b$ into batches of $\left(q^b?1\right)∕\left(q^c?1\right)$ subspaces of dimension $a$. In particular, this accounts for all numerically allowed subspace partition types $a^x{b}^y$ of $V\left(n,q\right)$ under some additional conditions, for example, when $e=b$.     

Group divisible 3‐designs with block size four and group type 1ns1

In this article, we mainly consider the existence problem of a group divisible design GDD $\left(3,4,n+s\right)$ of type $1^n{s}^1$. We present two recursive constructions for this configuration using candelabra systems and construct explicitly a few small examples admitting given automorphism groups. As an application, several new infinite classes of GDD $\left(3,4,n+s\right)$s of type $1^n{s}^1$ are produced. Meanwhile a few new infinite families on candelabra quadruple systems with group sizes being odd and stem size greater than one are also obtained.     

Deza graphs with parameters (v,k,k−2,a)

A Deza graph with parameters $\left(v,k,b,a\right)$ is a $k$‐regular graph on $v$ vertices in which the number of common neighbors of two distinct vertices takes one of the following values: $b$ or $a$, where $b\ge a$. In the previous papers Deza graphs with $b=k?1$ were characterized. In this paper, we characterize Deza graphs with $b=k?2$.     

On t‐designs and s‐resolvable t‐designs from hyperovals

Hyperovals in projective planes turn out to have a link with t‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t‐designs and s‐resolvable t‐designs from hyperovals in projective planes of order $2^n$. We prove that the construction works for $t\le 5$. In particular, for $t=5$ the construction yields a family of 5‐ $\left(2^n+2,8,70\left(2^{n?2}?1\right)\right)$ designs. For $t=4$ numerous infinite families of 4‐designs on $2^n+2$ points with block size $2k$ can be constructed for any $k\ge 4$. The construction assumes the existence of a 4‐ $\left(2^{n?1}+1,k,\lambda \right)$ design, called the indexing design, including the complete 4‐ $\left(2^{n?1}+1,k,\left(\genfrac{}{}{0ex}{}{2^{n?1}?3}{k?4}\right)\right)$ design. Moreover, we prove that if the indexing design is s‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s‐resolvable for $s=2,3$. We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals.     

The number of the non‐full‐rank Steiner triple systems

The $p$‐rank of a Steiner triple system (STS) $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF $\left(p\right)$. We derive a formula for the number of different STSs of order $v$ and given $2$‐rank $r_2$, $r_2<v$, and a formula for the number of STSs of order $v$ and given $3$‐rank $r_3$, $r_3<v?1$. Also, we prove that there are no STSs of $2$‐rank smaller than $v$ and, at the same time, $3$‐rank smaller than $v?1$. Our results extend previous study on enumerating STSs according to the rank of their codes, mainly by Tonchev, V.A. Zinoviev, and D.V. Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.     

New lower bounds on the size of (n,r)‐arcs in PG(2,q)

An $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ is a set of $n$ points such that each line contains at most $r$ of the selected points. It is well known that $\left(n,r\right)$‐arcs in $\text{PG}\left(2,q\right)$ correspond to projective linear codes. Let $m_r\left(2,q\right)$ denote the maximal number $n$ of points for which an $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ exists. In this paper we obtain improved lower bounds on $m_r\left(2,q\right)$ by explicitly constructing $\left(n,r\right)$‐arcs. Some of the constructed $\left(n,r\right)$‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.     

Design theory and some forbidden configurations

In this paper we relate $t$‐designs to a forbidden configuration problem in extremal set theory. Let ${\mathbf{1}}_t{\mathbf{0}}_{?}$ denote a column of $t$ 1's on top of $?$ 0's. Let $q?{\mathbf{1}}_t{\mathbf{0}}_{?}$ denote the $\left(t+?\right)\times q$ matrix consisting of $t$ rows of $q$ 1's and $?$ rows of $q$ 0's. We consider extremal problems for matrices avoiding certain submatrices. Let $A$ be a (0, 1)‐matrix forbidding any $\left(t+?\right)\times \left(\lambda +2\right)$ submatrix $\left(\lambda +2\right)?{\mathbf{1}}_t{\mathbf{0}}_{?}$. Assume $A$ is $m$‐rowed and only columns of sum $t+1,t+2,\text{…},m??$ are allowed to be repeated. Assume that $A$ has the maximum number of columns subject to the given restrictions. Assume $m$ is sufficiently large. Then $A$ has each column of sum $0,1,\text{…},t$ and $m??+1,m??+2,\text{…},m$ exactly once and, given the appropriate divisibility condition, the columns of sum $t+1$ correspond to a $t$‐design with block size $t+1$ and parameter $\lambda$. The proof derives a basic upper bound on the number of columns of $A$ by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design‐based constructions. We extend in a few directions.     

Colourings of star systems

An $e$‐star is a complete bipartite graph $K_{1,e}$. An $e$‐star system of order $n>1,S_e\left(n\right)$, is a partition of the edges of the complete graph $K_n$ into $e$‐stars. An $e$‐star system is said to be $k$‐colourable if its vertex set can be partitioned into $k$ sets (called colour classes) such that no $e$‐star is monochromatic. The system $S_e\left(n\right)$ is $k$‐chromatic if $S_e\left(n\right)$ is $k$‐colourable but is not $\left(k?1\right)$‐colourable. If every $k$‐colouring of an $e$‐star system can be obtained from some $k$‐colouring $?$ by a permutation of the colours, we say that the system is uniquely $k$‐colourable. In this paper, we first show that for any integer $k?2$, there exists a $k$‐chromatic 3‐star system of order $n$ for all sufficiently large admissible $n$. Next, we generalize this result for $e$‐star systems for any $e?3$. We show that for all $k?2$ and $e?3$, there exists a $k$‐chromatic $e$‐star system of order $n$ for all sufficiently large $n$ such that $n\equiv 0,1$ (mod $2e$). Finally, we prove that for all $k?2$ and $e?3$, there exists a uniquely $k$‐chromatic $e$‐star system of order $n$ for all sufficiently large $n$ such that $n\equiv 0,1$ (mod $2e$).     

On the structure of small strength‐2 covering arrays

A covering array $\text{CA}\left(N;t,k,v\right)$ of strength $t$ is an $N\times k$ array of symbols from an alphabet of size $v$ such that in every $N\times t$ subarray, every $t$‐tuple occurs in at least one row. A covering array is optimal if it has the smallest possible $N$ for given $t$, $k$, and $v$, and uniform if every symbol occurs $?N∕v?$ or $?N∕v?$ times in every column. Before this paper, the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erd?s‐Ko‐Rado Theorem, and 11 other parameter sets with $v>2$ and $N>v^2$. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper, a new lower bound as well as structural constraints for small uniform strength‐2 covering arrays is given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength‐2 covering array with $v>2$ and $N>v^2$ is now known for 21 parameter sets. Our constructive results continue to support the conjecture.     

Some new group divisible designs with block size 4 and two or three group sizes

Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely $2^3{5}^4,3^5{6}^2$, and $2^2{5}^5$. In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types ${\left(3m\right)}^4{\left(6m\right)}^q{\left(3n\right)}^1$ for $0\le n\le \left(q+1\right)m$ where $q\in \left\{2,3\right\}$ and solutions for 4‐GDDs of types $3^t{6}^s$ for a wide range of values of $s\ge 2,t\ge 4$ satisfying $t\equiv 0$ or $1\left(\mathrm{mod}4\right)$, including all cases with $4\le t\le s?1$. Most of the remaining unknown 4‐GDDs of type $3^t{6}^s$ have $\left(s?1\right)<t<2\left(s?1\right)$.     

Using block designs in crossing number bounds

The crossing number $\mathrm{\text{CR}}\left(G\right)$ of a graph $G=\left(V,E\right)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$‐planar crossing number of $G,{\mathrm{\text{CR}}}_k\left(G\right)$, is defined as the minimum of $\text{CR}\left(G_1\right)+\text{CR}\left(G_2\right)+?+\text{CR}\left(G_k\right)$ over all graphs $G_1,G_2,\dots ,G_k$ with ${\cup }_{i=1}^k{G}_i=G$. Pach et al [Comput. Geom.: Theory Appl. 68 (2018), pp. 2–6] showed that for every $k\ge 1$, we have ${\text{CR}}_k\left(G\right)\le (2/k^2?1/k^3)\text{CR}\left(G\right)$ and that this bound does not remain true if we replace the constant $2/k^2?1/k^3$ by any number smaller than $1/k^2$. We improve the upper bound to $(1/k^2)\left(1+o\left(1\right)\right)$ as $k\to \infty$. For the class of bipartite graphs, we show that the best constant is exactly $1/k^2$ for every $k$. The results extend to the rectilinear variant of the $k$‐planar crossing number.     

An extension of a construction of covering arrays

By Raaphorst et al, for a prime power $q$, covering arrays (CAs) with strength 3 and index 1, defined over the alphabet ${\mathbb{F}}_q$, were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in ${\mathbb{F}}_q\left[x\right]$. These arrays have $2q^3?1$ rows and $q^2+q+1$ columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed produced; and analyze the parameters of the generated arrays. Besides arrays that match the parameters of those of Raaphorst et al, we obtain arrays matching some constructions that use Chateauneuf‐Kreher doubling; in both cases these are some of the best arrays currently known for certain parameters.