arXiv subjects

Stefano Della Fiore

Publications and source records attributed to Stefano Della Fiore.

31 records · Page 2Linked to original sources

arXiv2023

Bounds on the Higher Degree Erdős-Ginzburg-Ziv Constants over $\mathbb{F}_q^n$

The classical Erdős-Ginzburg-Ziv constant of a group $G$ denotes the smallest positive integer $\ell$ such that any sequence $S$ of length at least $\ell$ contains a zero-sum subsequence of length $\exp(G)$. In a recent paper, Caro and Schmitt generalized this concept, using the $m$-th degree symmetric polynomial $e_m(S)$ instead of the sum of the elements of $S$ and considering subsequences of a given length $t$. In particular, they defined the higher degree Erdős-Ginzburg-Ziv constants $EGZ(t,R,m)$ of a finite commutative ring $R$ and presented several lower and upper bounds to these constants. This paper aims to provide lower and upper bounds for $EGZ(t,R,m)$ in case $R=\mathbb{F}_q^{n}$. The lower bounds here presented have been obtained, respectively, using Lovász Local Lemma and the Expurgation method and, for sufficiently large $n$, they beat the lower bound provided by Caro and Schmitt for the same kind of rings. Finally, we prove closed form upper bounds derived from the Ellenberg-Gijswijt and Sauermann results for the cap-set problem assuming that $q = p^k$, $t = p$, and $m=p-1$. Moreover, using the Slice Rank method we derive a convex optimization problem that provides the best bounds for $q = 3^k$, $t = 3$, $m=2$ and $k=2,3,4,5$.

math.CO↗

arXiv2023

Bounds and Algorithms for Frameproof Codes and Related Combinatorial Structures

In this paper, we study upper bounds on the minimum length of frameproof codes introduced by Boneh and Shaw to protect copyrighted materials. A $q$-ary $(k,n)$-frameproof code of length $t$ is a $t \times n$ matrix having entries in $\{0,1,\ldots, q-1\}$ and with the property that for any column $\mathbf{c}$ and any other $k$ columns, there exists a row where the symbols of the $k$ columns are all different from the corresponding symbol (in the same row) of the column $\mathbf{c}$. In this paper, we show the existence of $q$-ary $(k,n)$-frameproof codes of length $t = O(\frac{k^2}{q} \log n)$ for $q \leq k$, using the Lovász Local Lemma, and of length $t = O(\frac{k}{\log(q/k)}\log(n/k))$ for $q > k$ using the expurgation method. Remarkably, for the practical case of $q \leq k$ our findings give codes whose length almost matches the lower bound $Ω(\frac{k^2}{q\log k} \log n)$ on the length of any $q$-ary $(k,n)$-frameproof code and, more importantly, allow us to derive an algorithm of complexity $O(t n^2)$ for the construction of such codes.

cs.IT↗

arXiv2023

Sequencings in Semidirect Products via the Polynomial Method

The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j = \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different then ${\mathbf x}$ is a linear sequencing and if the partial sums are all different when $|i-j| \leq t$ then ${\mathbf x}$ is a $t$-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order $k$ of the non-identity elements of the dihedral group of order $2m$ has a linear sequencing when $k \leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than $k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$ of a non-abelian group of order three times a prime has a linear sequencing when $5 < k \leq 10$, unless $s_k$ is unavoidably the identity; and that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \leq 3$ and $t \leq 6$.

math.CO↗

arXiv2022

Variations on the Erdős distinct-sums problem

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erdős states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known to date is of the form $a_n>c\cdot 2^n/\sqrt{n}$. In this paper, we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to $λn$ be distinct. For this case, we derive lower and upper bounds on the smallest possible value of $a_n$.

math.CO↗

arXiv2022

Existence of $λ$-Fold Non-zero sum Heffter arrays through local considerations

In [12] was introduced, for cyclic groups, the class of partially filled arrays of the non-zero sum Heffter array that are, as the Heffter arrays, related to difference families, graph decompositions, and biembeddings. Here we generalize this definition to any finite groups. Given a subgroup $J$ of order $t$ of a group $G$, a $λ$-fold non-zero sum Heffter array over $G$ relative to $J$, $^λ\mathrm{N}\mathrm{H}_t(m,n; h,k)$, is an $m \times n$ p. f. array with entries in $G$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x\in G\setminus J$, the sum of the occurrence of $x$ and $-x$ is $λ$; the sum of the elements in every row and column is, following the natural orderings from left to right for the rows and from top to bottom for the columns, different from $0$ (in $G$). In [12], there was presented a complete, probabilistic, solution for the existence problem in case $λ=1$ and $G=\mathbb{Z}_v$ that is the starting point of this investigation. In this paper, we will consider the existence problem for a generic value of $λ$ and a generic finite group $G$, and we present an almost complete solution to this problem. In particular, we will prove, through local considerations (inspired by Lovász Local Lemma), that there exists a $λ$-fold non-zero sum Heffter array over $G$ relative to $J$ whenever the trivial necessary conditions are satisfied and $|G|=v\geq 41$. This value can be turned down to $29$ in case the array does not contain empty cells. Finally, we will show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide new infinite families of such embeddings.

math.CO↗

arXiv2022

Achievable Rates and Algorithms for Group Testing with Runlength Constraints

In this paper, we study bounds on the minimum length of $(k,n,d)$-superimposed codes introduced by Agarwal et al. [1], in the context of Non-Adaptive Group Testing algorithms with runlength constraints. A $(k,n,d)$-superimposed code of length $t$ is a $t \times n$ binary matrix such that any two 1's in each column are separated by a run of at least $d$ 0's, and such that for any column $\mathbf{c}$ and any other $k-1$ columns, there exists a row where $\mathbf{c}$ has $1$ and all the remaining $k-1$ columns have $0$. Agarwal et al. proved the existence of such codes with $t=Θ(dk\log(n/k)+k^2\log(n/k))$. Here we investigate more in detail the coefficients in front of these two main terms as well as the role of lower order terms. We show that improvements can be obtained over the construction in [1] by using different constructions and by an appropriate exploitation of the Lovász Local Lemma in this context. Our findings also suggest $O(n^k)$ randomized Las Vegas algorithms for the construction of such codes. We also extend our results to Two-Stage Group Testing algorithms with runlength constraints.

cs.IT↗

arXiv2022

Weak Sequenceability in Cyclic Groups

A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_i$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $s_k = s_0 = 0$. In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized in $[4]$ into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set $A$ do not sum to $0$ then there exists a simple path $P$ in the Cayley graph $Cay[G:\pm A]$ such that $Δ(P) = \pm A$. In this paper, inspired by this graph-theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk $W$ of girth bigger than $t$ (for a given $t < k$) and such that $Δ(W) = \pm A$. This is possible given that the partial sums $s_i$ and $s_j$ are different whenever $i$ and $j$ are distinct and $|i-j|\leq t$. In this case, we say that the set $A$ is $t$-weak sequenceable. The main result here presented is that any subset $A$ of $\mathbb{Z}_p\setminus \{0\}$ is $t$-weak sequenceable whenever $t<7$ or when $A$ does not contain pairs of type $\{x,-x\}$ and $t<8$.

math.CO↗

arXiv2022

On Sequences in Cyclic Groups with Distinct Partial Sums

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$. We demonstrate the sequenceability of subsets of size $k$ of $\mathbb{Z}_n \setminus \{ 0 \}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$. We obtain similar, but partial, results for $13 \leq k \leq 15$. This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable.

math.CO↗

arXiv2022

Improved Bounds for $(b,k)$-hashing

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists. Equivalently, determining the asymptotic growth of a largest subset of $\{1,2,\ldots,b\}^n$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general $b, k$, was derived by Fredman and Komlós in the '80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for the general $b,k$ case by Guruswami and Riazanov while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Costa and Dalai. In this paper, we both show how some of the latter results extend to $b\neq k$ and further strengthen the bounds for some specific small values of $b$ and $k$. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity which could be reduced by using more sophisticated and optimized algorithmic approaches.

math.CO↗

arXiv2022

Non-zero sum Heffter arrays and their applications

In this paper we introduce a new class of partially filled arrays that, as Heffter arrays, are related to difference families, graph decompositions and biembeddings. A non-zero sum Heffter array $\mathrm{N}\mathrm{H}(m,n; h,k)$ is an $m \times n$ p. f. array with entries in $\mathbb{Z}_{2nk+1}$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x\in \mathbb{Z}_{2nk+1}\setminus\{0\}$, either $x$ or $-x$ appears in the array; the sum of the elements in every row and column is different from $0$ (in $\mathbb{Z}_{2nk+1}$). Here first we explain the connections with relative difference families and with path decompositions of the complete multipartite graph. Then we present a complete solution for the existence problem and a constructive complete solution for the square case and for the rectangular case with no empty cells when the additional, very restrictive, property of "globally simple" is required. Finally, we show how these arrays can be used to construct biembeddings of complete graphs.

math.CO↗

arXiv2022

The maximum cardinality of trifferent codes with lengths 5 and 6

A code $\mathcal{C} \subseteq \{0, 1, 2\}^n$ is said to be trifferent with length $n$ when for any three distinct elements of $\mathcal{C}$ there exists a coordinate in which they all differ. Defining $\mathcal{T}(n)$ as the maximum cardinality of trifferent codes with length $n$, $\mathcal{T}(n)$ is unknown for $n \ge 5$. In this note, we use an optimized search algorithm to show that $\mathcal{T}(5) = 10$ and $\mathcal{T}(6) = 13$.

math.CO↗

arXiv2021

A note on $\overline{2}$-separable codes and $B_2$ codes

We derive a simple proof, based on information theoretic inequalities, of an upper bound on the largest rates of $q$-ary $\overline{2}$-separable codes that improves recent results of Wang for any $q\geq 13$. For the case $q=2$, we recover a result of Lindström, but with a much simpler derivation. The method easily extends to give bounds on $B_2$ codes which, although not improving on Wang's results, use much simpler tools and might be useful for future applications.

math.CO↗

arXiv2021

New upper bounds for $(b,k)$-hashing

For fixed integers $b\geq k$, the problem of perfect $(b,k)$-hashing asks for the asymptotic growth of largest subsets of $\{1,2,\ldots,b\}^n$ such that for any $k$ distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general $b, k$, was derived by Fredman and Komlós in the '80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for the general $b,k$ case by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Costa and Dalai. In this paper, we both show how some of the latter results extend to $b\neq k$ and further strengthen the bounds for some specific small values of $b$ and $k$. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity.

cs.IT↗