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Documents authored by He, Yuan

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He, Yuan

Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2017.56
Brief Announcement: Compact Topology of Shared-Memory Adversaries

Authors: Petr Kuznetsov, Thibault Rieutord, and Yuan He

Published in: LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

Abstract
The paper proposes a simple topological characterization of a large class of adversarial distributed-computing models via affine tasks: sub-complexes of the second iteration of the standard chromatic subdivision. We show that the task computability of a model in the class is precisely captured by iterations of the corresponding affine task. While an adversary is in general defined as a non-compact set of infinite runs, its affine task is just a finite subset of runs of the 2-round iterated immediate snapshot (IIS) model. Our results generalize and improve all previously derived topological characterizations of distributed-computing models.
Cite as

Petr Kuznetsov, Thibault Rieutord, and Yuan He. Brief Announcement: Compact Topology of Shared-Memory Adversaries. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 56:1-56:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kuznetsov_et_al:LIPIcs.DISC.2017.56,
  author =	{Kuznetsov, Petr and Rieutord, Thibault and He, Yuan},
  title =	{{Brief Announcement: Compact Topology of Shared-Memory Adversaries}},
  booktitle =	{31st International Symposium on Distributed Computing (DISC 2017)},
  pages =	{56:1--56:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-053-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{91},
  editor =	{Richa, Andr\'{e}a},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.56},
  URN =		{urn:nbn:de:0030-drops-80108},
  doi =		{10.4230/LIPIcs.DISC.2017.56},
  annote =	{Keywords: Adversarial models, Affine tasks, Topological characterization}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2016.6
Read-Write Memory and k-Set Consensus as an Affine Task

Authors: Eli Gafni, Yuan He, Petr Kuznetsov, and Thibault Rieutord

Published in: LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

Abstract
The wait-free read-write memory model has been characterized as an iterated Immediate Snapshot (IS) task. The IS task is affine — it can be defined as a (sub)set of simplices of the standard chromatic subdivision. In this paper, we highlight the phenomenon of a "natural" model that can be captured by an iterated affine task and, thus, by a subset of runs of the iterated immediate snapshot model. We show that the read-write memory model in which, additionally, k-set-consensus objects can be used is "natural" by presenting the corresponding simple affine task captured by a subset of 2-round IS runs. As an "unnatural" example, the model using the abstraction of Weak Symmetry Breaking (WSB) cannot be captured by a set of IS runs and, thus, cannot be represented as an affine task. Our results imply the first combinatorial characterization of models equipped with abstractions other than read-write memory that applies to generic tasks.
Cite as

Eli Gafni, Yuan He, Petr Kuznetsov, and Thibault Rieutord. Read-Write Memory and k-Set Consensus as an Affine Task. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gafni_et_al:LIPIcs.OPODIS.2016.6,
  author =	{Gafni, Eli and He, Yuan and Kuznetsov, Petr and Rieutord, Thibault},
  title =	{{Read-Write Memory and k-Set Consensus as an Affine Task}},
  booktitle =	{20th International Conference on Principles of Distributed Systems (OPODIS 2016)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-031-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{70},
  editor =	{Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.6},
  URN =		{urn:nbn:de:0030-drops-70759},
  doi =		{10.4230/LIPIcs.OPODIS.2016.6},
  annote =	{Keywords: iterated affine tasks, k-set consensus, k-concurrency, simplicial complexes, immediate snapshot}
}

Yuan, Chen

Document
DOI: 10.4230/LIPIcs.ITC.2023.1
Two-Round Perfectly Secure Message Transmission with Optimal Transmission Rate

Authors: Nicolas Resch and Chen Yuan

Published in: LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

Abstract
In the model of Perfectly Secure Message Transmission (PSMT), a sender Alice is connected to a receiver Bob via n parallel two-way channels, and Alice holds an 𝓁 symbol secret that she wishes to communicate to Bob. There is an unbounded adversary Eve that controls t of the channels, where n = 2t+1. Eve is able to corrupt any symbol sent through the channels she controls, and furthermore may attempt to infer Alice’s secret by observing the symbols sent through the channels she controls. The transmission is required to be (a) reliable, i.e., Bob must always be able to recover Alice’s secret, regardless of Eve’s corruptions; and (b) private, i.e., Eve may not learn anything about Alice’s secret. We focus on the two-round model, where Bob is permitted to first transmit to Alice, and then Alice responds to Bob. In this work we provide upper and lower bounds for the PSMT model when the length of the communicated secret 𝓁 is asymptotically large. Specifically, we first construct a protocol that allows Alice to communicate an 𝓁 symbol secret to Bob by transmitting at most 2(1+o_{𝓁→∞}(1))n𝓁 symbols. Under a reasonable assumption (which is satisfied by all known efficient two-round PSMT protocols), we complement this with a lower bound showing that 2n𝓁 symbols are necessary for Alice to privately and reliably communicate her secret. This provides strong evidence that our construction is optimal (even up to the leading constant).
Cite as

Nicolas Resch and Chen Yuan. Two-Round Perfectly Secure Message Transmission with Optimal Transmission Rate. In 4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{resch_et_al:LIPIcs.ITC.2023.1,
  author =	{Resch, Nicolas and Yuan, Chen},
  title =	{{Two-Round Perfectly Secure Message Transmission with Optimal Transmission Rate}},
  booktitle =	{4th Conference on Information-Theoretic Cryptography (ITC 2023)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-271-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{267},
  editor =	{Chung, Kai-Min},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.1},
  URN =		{urn:nbn:de:0030-drops-183297},
  doi =		{10.4230/LIPIcs.ITC.2023.1},
  annote =	{Keywords: Secure transmission, Information theoretical secure, MDS codes}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.89
List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2

Authors: Shu Liu, Chaoping Xing, and Chen Yuan

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Despite numerous results about the list decoding of Hamming-metric codes, development of list decoding on rank-metric codes is not as rapid as its counterpart. The bound of list decoding obeys the Gilbert-Varshamov bound in both the metrics. In the case of the Hamming-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and alphabet size, while in the case of the rank-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and column-to-row ratio (i.e., the ratio between the numbers of columns and rows). Hence, alphabet size and column-to-row ratio play a similar role for list decodability in each metric. In the case of the Hamming-metric, it is more challenging to list decode codes over smaller alphabets. In contrast, in the case of the rank-metric, it is more difficult to list decode codes with large column-to-row ratio. In particular, it is extremely difficult to list decode square matrix rank-metric codes (i.e., the column-to-row ratio is equal to 1). The main purpose of this paper is to explicitly construct a class of rank-metric codes 𝒞 of rate R with the column-to-row ratio up to 2/3 and efficiently list decode these codes with decoding radius beyond the decoding radius (1-R)/2 (note that (1-R)/2 is at least half of relative minimum distance δ). In literature, the largest column-to-row ratio of rank-metric codes that can be efficiently list decoded beyond half of minimum distance is 1/2. Thus, it is greatly desired to efficiently design list decoding algorithms for rank-metric codes with the column-to-row ratio bigger than 1/2 or even close to 1. Our key idea is to compress an element of the field F_qⁿ into a smaller F_q-subspace via a linearized polynomial. Thus, the column-to-row ratio gets increased at the price of reducing the code rate. Our result shows that the compression technique is powerful and it has not been employed in the topic of list decoding of both the Hamming and rank metrics. Apart from the above algebraic technique, we follow some standard techniques to prune down the list. The algebraic idea enables us to pin down the message into a structured subspace of dimension linear in the number n of columns. This "periodic" structure allows us to pre-encode the message to prune down the list.
Cite as

Shu Liu, Chaoping Xing, and Chen Yuan. List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{liu_et_al:LIPIcs.ICALP.2023.89,
  author =	{Liu, Shu and Xing, Chaoping and Yuan, Chen},
  title =	{{List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{89:1--89:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.89},
  URN =		{urn:nbn:de:0030-drops-181416},
  doi =		{10.4230/LIPIcs.ICALP.2023.89},
  annote =	{Keywords: Coding theory, List-decoding, Rank-metric codes}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.99
Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Authors: Nicolas Resch, Chen Yuan, and Yihan Zhang

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
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,L)_q-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 (p,𝓁,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_*+ε fraction of errors must have size O_ε(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_*-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.
Cite as

Nicolas Resch, Chen Yuan, and Yihan Zhang. Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 99:1-99:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{resch_et_al:LIPIcs.ICALP.2023.99,
  author =	{Resch, Nicolas and Yuan, Chen and Zhang, Yihan},
  title =	{{Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{99:1--99:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.99},
  URN =		{urn:nbn:de:0030-drops-181518},
  doi =		{10.4230/LIPIcs.ICALP.2023.99},
  annote =	{Keywords: Coding theory, List-decoding, List-recovery, Zero-rate thresholds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.104
Threshold Rates of Code Ensembles: Linear Is Best

Authors: Nicolas Resch and Chen Yuan

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
In this work, we prove new results concerning the combinatorial properties of random linear codes. By applying the thresholds framework from Mosheiff et al. (FOCS 2020) we derive fine-grained results concerning the list-decodability and -recoverability of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over 𝔽_q ε-close to capacity to list-recover with error radius ρ and input lists of size 𝓁. We show that the list-size L must be at least {log_q binom{q,𝓁}}-R}/ε, where R is the rate of the random linear code. This is analogous to a lower bound for list-decoding that was recently obtained by Guruswami et al. (IEEE TIT 2021B). As a comparison, we also pin down the list size of random codes which is {log_q binom{q,𝓁}}/ε. This result almost closes the O({q log L}/L) gap left by Guruswami et al. (IEEE TIT 2021A). This leaves open the possibility (that we consider likely) that random linear codes perform better than the random codes for list-recoverability, which is in contrast to a recent gap shown for the case of list-recovery from erasures (Guruswami et al., IEEE TIT 2021B). Next, we consider list-decoding with constant list-sizes. Specifically, we obtain new lower bounds on the rate required for: - List-of-3 decodability of random linear codes over 𝔽₂; - List-of-2 decodability of random linear codes over 𝔽_q (for any q). This expands upon Guruswami et al. (IEEE TIT 2021A) which only studied list-of-2 decodability of random linear codes over 𝔽₂. Further, in both cases we are able to show that the rate is larger than that which is possible for uniformly random codes. A conclusion that we draw from our work is that, for many combinatorial properties of interest, random linear codes actually perform better than uniformly random codes, in contrast to the apparently standard intuition that uniformly random codes are best.
Cite as

Nicolas Resch and Chen Yuan. Threshold Rates of Code Ensembles: Linear Is Best. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 104:1-104:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{resch_et_al:LIPIcs.ICALP.2022.104,
  author =	{Resch, Nicolas and Yuan, Chen},
  title =	{{Threshold Rates of Code Ensembles: Linear Is Best}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{104:1--104:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.104},
  URN =		{urn:nbn:de:0030-drops-164456},
  doi =		{10.4230/LIPIcs.ICALP.2022.104},
  annote =	{Keywords: Random Linear Codes, List-Decoding, List-Recovery, Threshold Rates}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.98
Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph

Authors: Chaoping Xing and Chen Yuan

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Locally recoverable codes are a class of block codes with an additional property called locality. A locally recoverable code with locality r can recover a symbol by reading at most r other symbols. Recently, it was discovered by several authors that a q-ary optimal locally recoverable code, i.e., a locally recoverable code achieving the Singleton-type bound, can have length much bigger than q+1. In this paper, we present both the upper bound and the lower bound on the length of optimal locally recoverable codes. Our lower bound improves the best known result in [Yuan Luo et al., 2018] for all distance d >= 7. This result is built on the observation of the parity-check matrix equipped with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure produces an optimal locally recoverable code if it satisfies a certain expansion property for subsets of F_q. To our surprise, this expansion property is then shown to be equivalent to a well-studied problem in extremal graph theory. Our upper bound is derived by an refined analysis of the arguments of Theorem 3.3 in [Venkatesan Guruswami et al., 2018].
Cite as

Chaoping Xing and Chen Yuan. Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 98:1-98:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{xing_et_al:LIPIcs.ICALP.2019.98,
  author =	{Xing, Chaoping and Yuan, Chen},
  title =	{{Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{98:1--98:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.98},
  URN =		{urn:nbn:de:0030-drops-106745},
  doi =		{10.4230/LIPIcs.ICALP.2019.98},
  annote =	{Keywords: Locally Repairable Codes, Hypergraph}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.41
How Long Can Optimal Locally Repairable Codes Be?

Authors: Venkatesan Guruswami, Chaoping Xing, and Chen Yuan

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case.
Cite as

Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. How Long Can Optimal Locally Repairable Codes Be?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 41:1-41:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2018.41,
  author =	{Guruswami, Venkatesan and Xing, Chaoping and Yuan, Chen},
  title =	{{How Long Can Optimal Locally Repairable Codes Be?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{41:1--41:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.41},
  URN =		{urn:nbn:de:0030-drops-94458},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.41},
  annote =	{Keywords: Locally Repairable Code, Singleton Bound}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.86
Subspace Designs Based on Algebraic Function Fields

Authors: Venkatesan Guruswami, Chaoping Xing, and Chen Yuan

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
Subspace designs are a (large) collection of high-dimensional subspaces {H_i} of F_q^m such that for any low-dimensional subspace W, only a small number of subspaces from the collection have non-trivial intersection with W; more precisely, the sum of dimensions of W cap H_i is at most some parameter L. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically q >= m). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over F^n for any field F, with logarithmic degree and expansion guarantee for subspaces of dimension Omega(n/(log(log(n)))).
Cite as

Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. Subspace Designs Based on Algebraic Function Fields. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{guruswami_et_al:LIPIcs.ICALP.2017.86,
  author =	{Guruswami, Venkatesan and Xing, Chaoping and Yuan, Chen},
  title =	{{Subspace Designs Based on Algebraic Function Fields}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{86:1--86:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.86},
  URN =		{urn:nbn:de:0030-drops-73712},
  doi =		{10.4230/LIPIcs.ICALP.2017.86},
  annote =	{Keywords: Subspace Design, Dimension Expander, List Decoding}
}
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