Explicit, almost optimal, epsilon-balanced codes

@article{TaShma2017ExplicitAO,
  title={Explicit, almost optimal, epsilon-balanced codes},
  author={Amnon Ta-Shma},
  journal={Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing},
  year={2017}
}
  • A. Ta-Shma
  • Published 19 June 2017
  • Computer Science
  • Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
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(k/ϵ2+o(1)). This improves upon all previous explicit constructions which were in the order of k2/ϵ2, k/ϵ3 or k5/4/ϵ5/2. The result is… 
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References

SHOWING 1-10 OF 32 REFERENCES
Constructing Small-Bias Sets from Algebraic-Geometric Codes
TLDR
An explicit construction of an $\eps$-biased set over $k$ bits of size with parameters nearly matching the lower bound is given, giving binary error correcting codes beating the Gilbert-Varshamov bound.
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
TLDR
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes, superior to previously known explicit construction in the zero-rate neighborhood.
Randomness Conductors and Constant-Degree Expansion Beyond the Degree / 2 Barrier
TLDR
The introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects, is introduced and it is shown that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above.
Simple Constructions of Almost k -wise Independent Random Variables
TLDR
Two of the constructions presented are based on bit sequences which are widely believed to posses “randomness properties” and can be viewed as an explanation and establishment of these beliefs.
Small-Bias Probability Spaces: Efficient Constructions and Applications
It is shown how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with “almost” equal probability. They are…
The PCP theorem by gap amplification
  • Irit Dinur
  • Mathematics, Computer Science
    STOC '06
  • 2006
TLDR
A new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system, is described.
Ramanujan Graphs
In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication…
A combinatorial construction of almost-ramanujan graphs using the zig-zag product
TLDR
A generalization of the zig-zag product that combines a large graph and several small graphs is proposed that gives a fully-explicit combinatorial construction of D-regular graphs having spectral gap 1-D-1/2 + o(1).
Undirected connectivity in log-space
TLDR
A deterministic, log-space algorithm that solves st-connectivity in undirected graphs and implies a way to construct in log- space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph.
Class of constructive asymptotically good algebraic codes
  • J. Justesen
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1972
TLDR
A decoding procedure is given that corrects all errors guaranteed correctable by the asymptotic lower bound on d .
...
1
2
3
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