Beating the probabilistic lower bound on perfect hashing

For an integer $q\ge 2$, a perfect $q$-hash code $C$ is a block code over $[q]:=\{1,\ldots,q\}$ of length $n$ in which every subset $\{\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_q\}$ of $q$ elements is separated, i.e., there exists $i\in[n]$ such that $\{\mathrm{proj}_i(\mathbf{c}_1),\dots,\mathrm{proj}_i(\mathbf{c}_q)\}=[q]$, where $\mathrm{proj}_i(\mathbf{c}_j)$ denotes the $i$th position of $\mathbf{c}_j$. Finding the maximum size $M(n,q)$ of perfect $q$-hash codes of length $n$, for given $q$ and $n$, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotical behavior of this problem. More precisely speaking, we will focus on the quantity $R_q:=\limsup_{n\rightarrow\infty}\frac{\log_2 M(n,q)}n$. A well-known probabilistic argument shows an existence lower bound on $R_q$, namely $R_q\ge\frac1{q-1}\log_2\left(\frac1{1-q!/q^q}\right)$ \cite{FK,K86}. This is still the best-known lower bound till now except for the case $q=3$ for which K\"{o}rner and Matron \cite{KM} found that the concatenation technique could lead to perfect $3$-hash codes beating this the probabilistic lower bound. The improvement on the lower bound on $R_3$ was discovered in 1988 and there has been no any progress on lower bound on $R_q$ for more than 30 years despite of some work on upper bounds on $R_q$. In this paper we show that this probabilistic lower bound can be improved for $q=4,8$ and all odd integers between $3$ and $25$, and \emph{all sufficiently large} $q$ with $q \pmod 4\neq 2$.Comment: arXiv admin note: text overlap with arXiv:1010.5764 by other author

Beating the probabilistic lower bound on perfect hashing

For an integer q > 2, a perfect q-hash code C is a block code over [q]:= {1,..., q} of length n in which every subset {c1, c2,..., cq} of q elements is separated, i.e., there exists i ∈ [n] such that {proji(c1),..., proji(cq)} = [q], where proji(cj) denotes the ith position of cj. Finding the maximum size M(n, q) of perfect q-hash codes of length n, for given q and n, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotical behavior of this problem. More precisely speaking, we will focus on the quantity Rq := li

New upper bounds for (b,k)(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 Koml\'os in the '80s and improved for certain $b\neq k$ by K\"orner 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.Comment: arXiv admin note: substantial text overlap with arXiv:2012.0062

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

In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of q ? 2. A code is called (p,L)_q-list-decodable if every radius pn Hamming ball contains less than L codewords; (p,?,L)_q-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length ? 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 (p,?,L)_q-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_*+? fraction of errors must have size O_?(1), 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_*-? 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

Further strengthening of upper bounds for perfect kk-Hashing

For a fixed integer $k$, 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 perfect $k$-hash family of $n$ functions exists. Equivalently, determining the asymptotic growth of a largest subset of $\{1,2,\ldots,k\}^n$ 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 $k$ was derived by Fredman and Koml\'os in the '80s. Only very recently this was improved for general $k$ by Guruswami and Riazanov while stronger results for small values of $k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Dalai and Costa. In this paper, we further improve the bounds for $5\leq k \leq 8$. 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

ZPERF: una familia de hash perfecto eficiente y de tamaño casi mínimo.

Implementación de una nueva y moderna forma de generar Familias de Hashes Perfectos en forma de Códigos Lineales, como se describe en un artículo escrito por los matemáticos Chaoping Xing y Chen Yuan, que poseen un rate menor con respecto a otros métodos parecidos. También se ha procesado el tiempo de cómputo de generación de estos códigos al igual que su rate de forma experimental, elementos que el artículo original no atacaba.<br /

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum