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Register a new userA note on the explicit constructions of tree codes over polylogarithmic-sized alphabet
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Prahladh Harsha
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudl{a}ks construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudl{a}k and Cohen-Haeupler-Schulmans constructions.
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An $(n,r,s)$-system is an $r$-uniform hypergraph on $n$ vertices such that every pair of edges has an intersection of size less than $s$. Using probabilistic arguments, R{o}dl and v{S}iv{n}ajov{a} showed that for all fixed integers $r> s ge 2$, there exists an $(n,r,s)$-system with independence number $Oleft(n^{1-delta+o(1)}right)$ for some optimal constant $delta >0$ only related to $r$ and $s$. We show that for certain pairs $(r,s)$ with $sle r/2$ there exists an explicit construction of an $(n,r,s)$-system with independence number $Oleft(n^{1-epsilon}right)$, where $epsilon > 0$ is an absolute constant only related to $r$ and $s$. Previously this was known only for $s>r/2$ by results of Chattopadhyay and Goodman
We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can achieve this only if the set $S$ has some very special algebraic structure, or if the degree $d$ is significantly smaller than $|S|$. We also give a near-linear time randomized algorithm, which is based on tools from list-decoding, to decode these codes from nearly half their minimum distance, provided $d < (1-epsilon)|S|$ for constant $epsilon > 0$. Our result gives an $m$-dimensional generalization of the well known decoding algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic version of the Schwartz-Zippel lemma.
Unitary $t$-designs are `good finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $tgeq 4$ if $dgeq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.
A quasi-Gray code of dimension $n$ and length $ell$ over an alphabet $Sigma$ is a sequence of distinct words $w_1,w_2,dots,w_ell$ from $Sigma^n$ such that any two consecutive words differ in at most $c$ coordinates, for some fixed constant $c>0$. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word $w_i$ into its successor $w_{i+1}$. We present construction of quasi-Gray codes of dimension $n$ and length $3^n$ over the ternary alphabet ${0,1,2}$ with worst-case read complexity $O(log n)$ and write complexity $2$. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension $n$ and length at least $2^n - 20n$ with worst-case read complexity $6+log n$ and write complexity $2$. This complements a recent result by Raskin [Raskin 17] who shows that any quasi-Gray code over binary alphabet of length $2^n$ has read complexity $Omega(n)$. Our results significantly improve on previously known constructions and for the odd-size alphabets we break the $Omega(n)$ worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. 14, Ben-Or and Cleve 92, Barrington 89, Coppersmith and Grossman 75].
For integers $n$ and $k$, the density Hales-Jewett number $c_{n,k}$ is defined as the maximal size of a subset of $[k]^n$ that contains no combinatorial line. We show that for $k ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to the maximal size of a cylinder intersection in the problem $Part_{n,k}$ of testing whether $k$ subsets of $[n]$ form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of $Part_{n,k}$, is equal to the minimal size of a partition of $[k]^n$ into subsets that do not contain a combinatorial line. Thus, the bound in cite{chattopadhyay2007languages} on $Part_{n,k}$ using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity. As a simple application we prove a lower bound on $c_{n,k}$, similar to the lower bound in cite{polymath2010moser} which is roughly $c_{n,k}/k^n ge exp(-O(log n)^{1/lceil log_2 krceil})$. This lower bound follows from a protocol for $Part_{n,k}$. It is interesting to better understand the communication complexity of $Part_{n,k}$ as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.
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