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… 
List-Decoding Random Walk XOR Codes Near the Johnson Bound
TLDR
The algorithm works for Ta-Shma’s original code, and it is able to list-decode almost all the way to the Johnson bound: it can recover from a 1−ρ 2 −fraction of errors as long as ρ ≥ 2 √ ε.
List-Decoding XOR Codes Near the Johnson Bound
TLDR
The algorithm works for Ta-Shma’s original code, and it is able to list-decode almost all the way to the Johnson bound: it can recover from a 1 − ρ 2 − fraction of errors as long as ρ ≥ 2 √ ε .
Linear Programming Bounds for Almost-Balanced Binary Codes
TLDR
An optimal solution to Delsarte’s LP is given for the almost-balanced case with large distance d ≥ (n−√n)/2+1, which shows that the optimal value of the LP coincides with the Grey-Rankin bound for self-complementary codes.
Near-linear time decoding of Ta-Shma’s codes via splittable regularity
TLDR
A new weak regularity decomposition is obtained for (possibly sparse) splittable collections W ⊆ [n]k, similar to the regularities decomposition for dense structures by Frieze and Kannan [FOCS 1996].
Punctured Low-Bias Codes Behave Like Random Linear Codes
TLDR
All current (and future) achievability bounds for list-decodability of random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) “mother” code, thus giving a derandomization of RLCs in the context of achieving Shannon capacity as well.
Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity
TLDR
All current (and future) list-decodability bounds shown for random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) code, which can be viewed as a general derandomization result applicable torandom linear codes.
Near-Optimal Cayley Expanders for Abelian Groups
TLDR
An efficient deterministic algorithm is given that outputs an expanding generating set for any finite abelian group and is an extension of the bias amplification technique of Ta-Shma, who used random walks on expanders to obtain expanding generating sets over the additive group of F2.
Randomness Efficient Noise Stability and Generalized Small Bias Sets
TLDR
A randomness efficient version of the linear noise operator Tρ from boolean function analysis is presented by constructing a sparse linear operator on the space of boolean functions → {0, 1} with similar eigenvalue profile to Tρ by constructing an explicitly constructible “sparse” noisy hypercube graph that is a small set expander.
List Decoding with Double Samplers
TLDR
The notion of "double Samplers", first introduced by Dinur and Kaufman, are developed, which are samplers with additional combinatorial properties, and whose existence is proved using high dimensional expanders, to show how double sampler give a generic way of amplifying distance in a way that enables efficient list-decoding.
A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding
TLDR
This work revisits a highly-successful online design technique, which was nonetheless under-utilized in the area of online matching, namely (lossless) online rounding of fractional algorithms, and proves a sharp threshold of (1 ± o(1)) log logn random bits for the randomness necessary and sufficient to outperform deterministic algorithms for this problem, as well as its vertex-weighted generalization.
...
...

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.
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.
A Chernoff Bound for Random Walks on Expander Graphs
  • D. Gillman
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1998
TLDR
The method of taking the sample average from one trajectory is a more efficient estimate of /spl pi/(A) than the standard method of generating independent sample points from several trajectories and improves the algorithms of Jerrum and Sinclair (1989) for approximating the number of perfect matchings in a dense graph.
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
TLDR
A decoding procedure is given that corrects all errors guaranteed correctable by the asymptotic lower bound on d .
...
...