相关论文: The Improvement of the Bound on Hash Family
In this paper, we present a new lower bound on the size of separating hash families of type {w_1^{q-1},w_2} where w_1 < w_2. Our result extends the paper by Guo et al. on binary frameproof codes. This bound compares well against known…
In this short note we give a new upper bound for the size of a set family with a single Hamming distance. Our proof is an application of the linear algebra bound method.
In this paper, we present a new construction for strong separating hash families by using hypergraphs and obtain some optimal separating hash families. We also improve some previously known bounds of separating hash families.
Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory,…
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new…
We present new upper bounds on the parameters of batch codes with restricted query size. These bounds are an improvement on the Singleton bound. The techniques for derivations of these bounds are based on the ideas in the literature for…
In this paper, we study $w$-frameproof codes, which are equivalent to $\{1,w\}$-separating hash families. Our main results concern binary codes, which are defined over an alphabet of two symbols. For all $w \geq 3$, and for $w+1 \leq N \leq…
Let $La(n,P)$ be the maximum size of a family of subsets of $[n]= \{1,2, ..., n \}$ not containing $P$ as a (weak) subposet, and let $h(P)$ be the length of a longest chain in $P$. The best known upper bound for $La(n,P)$ in terms of $|P|$…
We present new lower bounds for the size of perfect and separating hash families ensuring their existence. Such new bounds are based on the algorithmic Cluster expansion improved version of the Lov\'asz Local Lemma, which also implies that…
Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of universe for a $t$-perfect hash family of length $N$…
In this paper we consider the problem of deciding membership in Dyck languages, a fundamental family of context-free languages, comprised of well-balanced strings of parentheses. In this problem we are given a string of length $n$ in the…
Hashing has been widely used for efficient similarity search based on its query and storage efficiency. To obtain better precision, most studies focus on designing different objective functions with different constraints or penalty terms…
We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}^d under the Hamming distance. Recall that here H is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y in {0,1}^d with…
We propose to use the concept of the Hamming bound to derive the optimal criteria for learning hash codes with a deep network. In particular, when the number of binary hash codes (typically the number of image categories) and code length…
We say a subset $C \subseteq \{1,2,\dots,k\}^n$ is a $k$-hash code (also called $k$-separated) if for every subset of $k$ codewords from $C$, there exists a coordinate where all these codewords have distinct values. Understanding the…
Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming…
Separating codes have their applications in collusion-secure fingerprinting for generic digital data, while they are also related to the other structures including hash family, intersection code and group testing. In this paper we study…
The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the…
Let $K_n$ denote the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the number sequence $$ c_n=\min\{\lambda\mid\lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad…