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Given a $t$-$(v, k, lambda)$ design, $mathcal{D}=(X,mathcal{B})$, a zero-sum $n$-flow of $mathcal{D}$ is a map $f : mathcal{B}longrightarrow {pm1,ldots, pm(n-1)}$ such that for any point $xin X$, the sum of $f$ over all blocks incident with $x$ is zero. For a positive integer $k$, we find a zero-sum $k$-flow for an STS$(u w)$ and for an STS$(2v+7)$ for $vequiv 1~(mathrm{mod}~4)$, if there are STS$(u)$, STS$(w)$ and STS$(v)$ such that the STS$(u)$ and STS$(v)$ both have a zero-sum $k$-flow. In 2015, it was conjectured that for $v>7$ every STS$(v)$ admits a zero-sum $3$-flow. Here, it is shown that many cyclic STS$(v)$ have a zero-sum $3$-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.
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Let $X$ be a $v$-set, $B$ a set of 3-subsets (triples) of $X$, and $B^+cupB^-$ a partition of $B$ with $|B^-|=s$. The pair $(X,B)$ is called a simple signed Steiner triple system, denoted by ST$(v,s)$, if the number of occurrences of every 2-subset of $X$ in triples $BinB^+$ is one more than the number of occurrences in triples $BinB^-$. In this paper we prove that $st(v,s)$ exists if and only if $vequiv1,3pmod6$, $v e7$, and $sin{0,1,...,s_v-6,s_v-4,s_v}$, where $s_v=v(v-1)(v-3)/12$ and for $v=7$, $sin{0,2,3,5,6,8,14}$.
The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Omega subset B(G$), let $d_{Omega}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|geq t$ has a subsequence in $Omega$.We provide some first results and open problems on $d_{Omega}(G)$.
The $p$-rank of a Steiner triple system $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF$(p)$. We derive a formula for the number of different Steiner triple systems of order $v$ and given $2$-rank $r_2$, $r_2<v$, and a formula for the number of Steiner triple systems of order $v$ and given $3$-rank $r_3$, $r_3<v-1$. Also, we prove that there are no Steiner triple systems of $2$-rank smaller than $v$ and, at the same time, $3$-rank smaller than $v-1$. Our results extend previous work on enumerating Steiner triple systems 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.
Consider the equation $mathcal{E}: x_1+ cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $rmid k$. The number $S_{mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $chi: [1, t] to {0, 1}$ there exists a solution $(hat{x}_1, hat{x}_2, ldots, hat{x}_k)$ to the equation $mathcal{E}$ satisfying $displaystyle sum_{i=1}^kchi(hat{x}_i) equiv 0pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and show, more generally, that $S_{mathfrak{z}, 2}(k, r)=kr - 2r+1$ for all positive integers $k$ and $r$ with $k>r$ and $r mid k$.
In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of order $v$, where $v=3^k$, and $3$-rank $v-k$. We develop an approach to generalize this bound and estimate the number of isomorphism classes of STS$(v)$ of rank $v-k-1$ for an arbitrary $v$ of form $3^kT$.
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