skip to main content
10.1145/3055399.3055408acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
research-article

Explicit, almost optimal, epsilon-balanced codes

Published: 19 June 2017 Publication History

Abstract

The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance 1-ϵ/2 and rate close to the Gilbert-Varshamov bound, attracted a lot of attention in recent decades. In this paper we solve the problem almost optimally and show an explicit ϵ-biased set over k bits with support size O(k2+o(1)). This improves upon all previous explicit constructions which were in the order of k22, k3 or k5/45/2. The result is close to the Gilbert-Varshamov bound which is O(k2) and the lower bound which is Ω(k2 log1/ϵ).
The main technical tool we use is bias amplification with the s-wide replacement product. The sum of two independent samples from an ϵ-biased set is ϵ2 biased. Rozenman and Wigderson showed how to amplify the bias more economically by choosing two samples with an expander. Based on that they suggested a recursive construction that achieves sample size O(k4). We show that amplification with a long random walk over the s-wide replacement product reduces the bias almost optimally.

Supplementary Material

MP4 File (d3_sa_pm_t2.mp4)

References

[1]
Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. 2004. PRIMES is in P. Annals of mathematics (2004), 781–793.
[2]
N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. 1992. Construction of asymptotically good, low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory 38 (1992), 509–516.
[3]
N. Alon, O. Goldreich, J. Håstad, and R. Peralta. 1992. Simple Constructions of Almost k–wise Independent Random Variables. Random Structures and Algorithms 3, 3 (1992), 289–303.
[4]
N. Alon, A. Lubotzky, and A. Wigderson. 2001. Semi-direct product in groups and zig-zag product in graphs: connections and applications. In Proceedings of the 42nd FOCS. 630–637.
[5]
N. Alon, O. Schwartz, and A. Shapira. 2007.
[6]
An Elementary Construction of Constant Degree Expanders. In SODA. 454–458.
[7]
A. Ben-Aroya and A. Ta-Shma. 2011. A combinatorial construction of almost-Ramanujan graphs using the zig-zag product. SIAM J. Comput. 40, 2 (2011), 267–290.
[8]
A. Ben-Aroya and A. Ta-Shma. 2013.
[9]
Constructing Small-Bias Sets from Algebraic-Geometric Codes. Theory of Computing 9, 5 (2013), 253–272.
[10]
A. Bogdanov. 2012. A different way to improve the bias via expanders. Topics in (and out) the theory of computing, Lecture 12. (2012).
[11]
M. Capalbo, O. Reingold, S. Vadhan, and A. Wigderson. 2002.
[12]
Randomness Conductors and Constant-Degree Expansion Beyond the Degree / 2 Barrier. In STOC. 659–668.
[13]
I. Dinur. 2007.
[14]
The PCP theorem by gap amplification. Journal of the ACM (JACM) 54, 3 (2007), 12.
[15]
D. Gillman. 1998.
[16]
A Chernoff bound for random walks on expander graphs. SIAM J. Comput. 27, 4 (1998), 1203–1220.
[17]
S. Hoory, N. Linial, and A. Wigderson. 2006. Expander graphs and their applications. Bulletin of the AMS 43, 4 (2006), 439–561.
[18]
J. Justesen. 1972. Class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory 18, 5 (1972), 652–656.
[19]
A. Lubotzky, R. Philips, and P. Sarnak. 1988. Ramanujan Graphs. Combinatorica 8 (1988), 261–277.
[20]
J. Naor and M. Naor. 1993. Small–Bias Probability Spaces: Efficient Constructions and Applications. SIAM J. Comput. 22, 4 (1993), 838–856.
[21]
A. Nilli. 1991. On the second eigenvalue of a graph. Discrete Mathematics 91, 2 (1991), 207–210.
[22]
Anonymous Referee. 2009.
[23]
Reperee report. Private communication from the STOC 2009 committee. (2009).
[24]
O. Reingold. 2008.
[25]
Undirected connectivity in log-space. Journal of the ACM (JACM) 55, 4 (2008), 17.
[26]
O. Reingold, S. Vadhan, and A. Wigderson. 2002.
[27]
Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics 155, 1 (2002), 157–187.
[28]
E. Rozenman and S. Vadhan. 2005. Derandomized squaring of graphs. In RANDOM. 436–447.
[29]
V. Shoup. 1990. New algorithms for finding irreducible polynomials over finite fields. Math. Comp. 54, 189 (1990), 435–447.
[30]
M. Sudan. 2001.
[31]
Algorithmic introduction to coding theory, Lecture 6. http://people.csail.mit.edu/madhu/FT01/scribe/lect6.ps. (2001).
[32]
A. Ta-Shma. 2017.
[33]
Explicit, Almost optimal, epsilon Balanced Codes. Technical Report. ECCC TR17-041.

Cited By

View all
  • (2024)Linear Codes for Hyperdimensional ComputingNeural Computation10.1162/neco_a_0166536:6(1084-1120)Online publication date: 10-May-2024
  • (2024)Explicit Codes for Poly-Size Circuits and Functions That Are Hard to Sample on Low Entropy DistributionsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649735(2028-2038)Online publication date: 10-Jun-2024
  • (2024)Analyzing Ta-Shma’s Code via the Expander Mixing LemmaIEEE Transactions on Information Theory10.1109/TIT.2023.330461470:2(1040-1049)Online publication date: Feb-2024
  • Show More Cited By

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
June 2017
1268 pages
ISBN:9781450345286
DOI:10.1145/3055399
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 June 2017

Permissions

Request permissions for this article.

Check for updates

Badges

  • Best Paper

Author Tags

  1. Eps-bias
  2. Wide replacement product
  3. Zig-Zag product

Qualifiers

  • Research-article

Conference

STOC '17
Sponsor:
STOC '17: Symposium on Theory of Computing
June 19 - 23, 2017
Montreal, Canada

Acceptance Rates

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Upcoming Conference

STOC '25
57th Annual ACM Symposium on Theory of Computing (STOC 2025)
June 23 - 27, 2025
Prague , Czech Republic

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)52
  • Downloads (Last 6 weeks)4
Reflects downloads up to 20 Jan 2025

Other Metrics

Citations

Cited By

View all
  • (2024)Linear Codes for Hyperdimensional ComputingNeural Computation10.1162/neco_a_0166536:6(1084-1120)Online publication date: 10-May-2024
  • (2024)Explicit Codes for Poly-Size Circuits and Functions That Are Hard to Sample on Low Entropy DistributionsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649735(2028-2038)Online publication date: 10-Jun-2024
  • (2024)Analyzing Ta-Shma’s Code via the Expander Mixing LemmaIEEE Transactions on Information Theory10.1109/TIT.2023.330461470:2(1040-1049)Online publication date: Feb-2024
  • (2024)Tight Bounds for the Zig-Zag Product2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00094(1470-1499)Online publication date: 27-Oct-2024
  • (2024)Strong blocking sets and minimal codes from expander graphsTransactions of the American Mathematical Society10.1090/tran/9205Online publication date: 11-Jun-2024
  • (2024)Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching ProgramsAlgorithmica10.1007/s00453-024-01251-286:10(3153-3185)Online publication date: 29-Jul-2024
  • (2024)Pseudorandom Error-Correcting CodesAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68391-6_10(325-347)Online publication date: 18-Aug-2024
  • (2024)Traceable Secret Sharing: Strong Security and Efficient ConstructionsAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68388-6_9(221-256)Online publication date: 17-Aug-2024
  • (2024)Non-malleable Codes with Optimal Rate for Poly-Size CircuitsAdvances in Cryptology – EUROCRYPT 202410.1007/978-3-031-58737-5_2(33-54)Online publication date: 26-May-2024
  • (2023)Hardness against Linear Branching Programs and MoreProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.9(1-27)Online publication date: 17-Jul-2023
  • Show More Cited By

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media