10 research outputs found
Three Query Locally Decodable Codes with Higher Correctness Require Exponential Length
Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding,
possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length.
The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is for nonbinary codes, where an adversary is allowed to corrupt up to delta fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff, and it is below 1 - 3 delta.
We show that achieving slightly larger correctness (as a function of ) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our results hold for linear codes over arbitrary finite fields and for binary nonlinear codes.
Considering larger number of queries, we obtain lower bounds for q-query codes for q>3, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms.
In addition, our results imply tradeoffs on the parameters of error correcting data structures
Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity
A square matrix V is called rigid if every matrix V\u27 obtained by altering a small number of entries of has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field F_2.
Stated informally, we conjecture that under a suitable embedding of F_2^n into R^n, vectors arising from a low dimensional F_2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
On Embeddings of l_1^k from Locally Decodable Codes
We show that any -query locally decodable code (LDC) gives a copy of
with small distortion in the Banach space of -linear forms on
, provided and where , , and the distortion are simple functions of the code
parameters. We exhibit the copy of by constructing a basis for it
directly from "smooth" LDC decoders. Based on this, we give alternative proofs
for known lower bounds on the length of 2-query LDCs. Using similar techniques,
we reprove known lower bounds for larger . We also discuss the relation with
an alternative proof, due to Pisier, of a result of Naor, Regev, and the author
on cotype properties of projective tensor products of spaces
Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC
The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ? and, say, constant distance to an asymptotically good LCC with q^poly(1/?) queries.
Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance ? and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/? asymptotically good LDC. This improves upon the poly(1/?) overhead obtained by the AEL distance amplification procedure [Alon and Luby, 1996; Alon et al., 1995].
Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively
The method of multiplicities
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 93-98).Polynomials have played a fundamental role in the construction of objects with interesting combinatorial properties, such as error correcting codes, pseudorandom generators and randomness extractors. Somewhat strikingly, polynomials have also been found to be a powerful tool in the analysis of combinatorial parameters of objects that have some algebraic structure. This method of analysis has found applications in works on list-decoding of error correcting codes, constructions of randomness extractors, and in obtaining strong bounds for the size of Kakeya Sets. Remarkably, all these applications have relied on very simple and elementary properties of polynomials such as the sparsity of the zero sets of low degree polynomials. In this thesis we improve on several of the results mentioned above by a more powerful application of polynomials that takes into account the information contained in the derivatives of the polynomials. We call this technique the method of multiplicities. The derivative polynomials encode information about the high multiplicity zeroes of the original polynomial, and by taking into account this information, we are about to meaningfully reason about the zero sets of polynomials of degree much higher than the underlying field size. This freedom of using high degree polynomials allows us to obtain new and improved constructions of error correcting codes, and qualitatively improved analyses of Kakeya sets and randomness extractors.by Shubhangi Saraf.Ph.D
Any Errors in this Dissertation are Probably Fixable: Topics in Probability and Error Correcting Codes.
We study two problems in coding theory, list-decoding and local-decoding. We take a probabilistic approach to these problems, in contrast to more typical algebraic approaches.
In list-decoding, we settle two open problems about the list-decodability of some well-studied ensembles of codes. First, we show that random linear codes are optimally list-decodable, and second, we show that there exist Reed-Solomon codes which are (nearly) optimally list-decodable. Our approach uses high-dimensional probability. We extend this framework to apply to a large family of codes obtained through random operations.
In local-decoding, we use expander codes to construct locally-correctible linear codes with rate approaching 1. Until recently, such codes were conjectured not to exist, and before this work the only known constructions relied on algebraic, rather than probabilistic and combinatorial, methods.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108844/1/wootters_1.pd