Beating the probabilistic lower bound on perfect hashing

Xing, Chaoping; Yuan, Chen

doi:10.1137/1.9781611976465.3

Cited by 6 publications

(3 citation statements)

References 10 publications

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“…However, the latter bound log 3 3 2 remains the best known for q " 3 (called the trifference problem by Körner). For larger q, both lower [51] and upper bounds [2,11,27,10,12] can be improved. However, improving the bound for q " 3 is recognized as a formidable challenge.…”

Section: Discussion On Related Workmentioning

confidence: 99%

Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Resch,

Yuan,

Zhang

2024

IEEE Trans. Inform. Theory

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In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of q ě 2. A code is called pp, Lqq-list-decodable if every radius pn Hamming ball contains less than L codewords; pp, ℓ, Lqq-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length ℓ and again stipulate that there be less than L codewords.Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate pp, ℓ, Lqq-list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by p˚, we in fact show that codes correcting a p˚`ε fraction of errors must have size Oεp1q, i.e., independent of n. Such a result is typically referred to as a "Plotkin bound." To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a p˚´ε fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery.Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.

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Section: Discussion On Related Workmentioning

confidence: 99%

Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Resch,

Yuan,

Zhang

2024

IEEE Trans. Inform. Theory

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“…A perfect

q

‐ hash code of length

n

is a subset

C

\lbrace 0, 1, \dots, q - 1\rbrace ^n

such that for any

q

distinct elements in

C

, there is a coordinate where they have pairwise distinct values. Understanding the largest possible size of a perfect

q

‐hash code is a natural extremal problem that has gained much attention since the 1980s because of its connections to various topics in cryptography, information theory, and computer science [34, 40, 53, 54]. We will focus on the

q = 3

case where these codes are also known as trifferent codes , and the problem of determining their largest possible size is called the trifference problem .…”

Section: Introductionmentioning

confidence: 99%

Blocking sets, minimal codes and trifferent codes

Bishnoi,

D'haeseleer,

Gijswijt

et al. 2024

Journal of London Math Soc

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We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension‐2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on a strong blocking set. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length with the property that for any three distinct codewords there is a coordinate where they all have distinct values. Over the finite field , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length has size at most , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length and size at least , thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension‐2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to , we obtain linear trifferent codes of size at least , improving the current best explicit construction that has size .

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“…, q−1} n such that for any q distinct elements in C, there is a coordinate where they all have pairwise distinct values. Understanding the largest possible size of a perfect q-hash code is a natural extremal problem that has gained much attention since the 1980s because of its connections to various topics in cryptography, information theory, and computer science [30,36,47,48]. We will focus on the q = 3 case where these codes are also known as trifferent codes, and the problem of determining their largest possible size is called the trifference problem.…”

Section: Introductionmentioning

confidence: 99%

Blocking sets, minimal codes and trifferent codes

Bishnoi¹,

D'haeseleer²,

Gijswijt³

et al. 2023

Preprint

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We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set.Over the finite field F 3 , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n has size at most 3 n/4.55 , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n and size at least 1 3 (9/5) n/4 , thus (asymptotically) matching the best lower bound on trifferent codes.We also give explicit constructions of affine blocking sets with respect to codimension-2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to F 3 , we obtain linear trifferent codes of size at least 3 7n/240 , improving the current best explicit construction that has size 3 n/112 .

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Beating the probabilistic lower bound on perfect hashing

Cited by 6 publications

References 10 publications

Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Blocking sets, minimal codes and trifferent codes

Blocking sets, minimal codes and trifferent codes

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