Bounds for separating hash families

Bazrafshan, Marjan; Trung, Tran van

doi:10.1016/j.jcta.2010.11.006

Cited by 24 publications

(26 citation statements)

References 16 publications

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“…Therefore, during the recent years they have received remarkable attentions. Many efforts have been made to determine the upper and lower bounds for separating hash families, see for example, [5], [9], [26] and [31].…”

Section: Introductionmentioning

confidence: 99%

Some intriguing upper bounds for separating hash families

Shangguan

Wang

2018

Sci. China Math.

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An N × n matrix on q symbols is called {w 1 , . . . , w t }-separating if for arbitrary t pairwise disjoint column sets C 1 , . . . , C t with |C i | = w i for 1 ≤ i ≤ t, there exists a row f such that f (C 1 ), . . . , f (C t ) are also pairwise disjoint, where f (C i ) denotes the collection of components of C i restricted to row f . Given integers N, q and w 1 , . . . , w t , denote by C(N, q, {w 1 , . . . , w t }) the maximal n such that a corresponding matrix does exist. The determination of C(N, q, {w 1 , . . . , w t }) has received remarkable attentions during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w 1 , . . . , w t }). The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second one is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N, q, {w 1 , . . . , w t }), which significantly improve the previously known results.

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Section: Introductionmentioning

confidence: 99%

Some intriguing upper bounds for separating hash families

Shangguan

Wang

2018

Sci. China Math.

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“…with |A (1,2) i,j | ≥ 2 in the ith row of A, namely, the number of "⋆" in the ith row of B. Let ρ i = j∈Q,2≤|A…”

Section: An Upper Boundmentioning

confidence: 99%

“…It is easy to check that the forbidden configurations in (1) are also forbidden configurations of PHF (3; r 3 , r 2 , 3). Now, let us consider the forbidden configuration in (2).…”

Section: 1mentioning

confidence: 99%

Bounds and constructions for $${\overline{3}}$$ 3 ¯ -separable codes with length 3

Cheng

Jiang

et al. 2015

Des. Codes Cryptogr.

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Abstract. Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let C be a code of length n over an alphabet of q letters. The descendant code desc(C0) of C0 = {c1, c2, . . . , ct} ⊆ C is defined to be the set of words x = (x1, x2, . . . , xn)T such that xi ∈ {c1,i, c2,i, . . . , ct,i} for all i = 1, . . . , n, where cj = (cj,1, cj,2, . . . , cj,n)T . C is a t-separable code if for any two distinct C1, C2 ⊆ C with |C1| ≤ t, |C2| ≤ t, we always have desc(C1) = desc(C2). Let M (t, n, q) denote the maximal possible size of such a separable code. In this paper, an upper bound on M (3, 3, q) is derived by considering an optimization problem related to a partial Latin square, and then two constructions for 3-SC(3, M, q)s are provided by means of perfect hash families and Steiner triple systems.

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“…Fig. 3 gives an example of a heterogeneous DHF(10; 13, v, 5, 2) with v = (9, 9,9,3,3,3,3,4,5,2). We abbreviate repetition in such a tuple by using an exponential notation that indicates the repetition in the exponent v = (9 3 3 4 4 1 5 1 2 1 ).…”

Section: ↓ ↓ ↓mentioning

confidence: 99%

“…This yields a logarithmic lower bound. More precise bounds are known in many cases (for example, [3,6]), but all are Ω(log k) bounds on N. When d, v, and W are fixed; v d; and d max{s: {w 1 , . .…”

Section: ↓ ↓ ↓mentioning

confidence: 99%

Strengthening hash families and compressive sensing

Colbourn

Horsley

Syrotiuk

2012

Journal of Discrete Algorithms

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The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein-Lovász-Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived.

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Bounds for separating hash families

Abstract: This paper aims to present new upper bounds on the size of separating hash families. These bounds improve previously known bounds for separating hash families.

Cited by 24 publications

References 16 publications

Some intriguing upper bounds for separating hash families

Some intriguing upper bounds for separating hash families

Bounds and constructions for $${\overline{3}}$$ 3 ¯ -separable codes with length 3

Strengthening hash families and compressive sensing

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