1,130 research outputs found
Lower bounds for constant query affine-invariant LCCs and LTCs
Affine-invariant codes are codes whose coordinates form a vector space over a
finite field and which are invariant under affine transformations of the
coordinate space. They form a natural, well-studied class of codes; they
include popular codes such as Reed-Muller and Reed-Solomon. A particularly
appealing feature of affine-invariant codes is that they seem well-suited to
admit local correctors and testers.
In this work, we give lower bounds on the length of locally correctable and
locally testable affine-invariant codes with constant query complexity. We show
that if a code is an -query
locally correctable code (LCC), where is a finite field and
is a finite alphabet, then the number of codewords in is
at most . Also, we show that if
is an -query locally testable
code (LTC), then the number of codewords in is at most
. The dependence on in these
bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan
(ITCS `13) construct affine-invariant codes via lifting that have the same
asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas
previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive
similar results.
Our analysis uses higher-order Fourier analysis. In particular, we show that
the codewords corresponding to an affine-invariant LCC/LTC must be far from
each other with respect to Gowers norm of an appropriate order. This then
allows us to bound the number of codewords, using known decomposition theorems
which approximate any bounded function in terms of a finite number of
low-degree non-classical polynomials, upto a small error in the Gowers norm
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}
Quantum Locally Testable Codes
We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a
definition together with a simplification, denoted sLTCs, for the special case
of stabilizer codes, together with some basic results using those definitions.
The most crucial parameter of such codes is their soundness, ,
namely, the probability that a randomly chosen constraint is violated as a
function of the distance of a word from the code (, the relative
distance from the code, is called the proximity). We then proceed to study
limitations on qLTCs. In our first main result we prove a surprising,
inherently quantum, property of sLTCs: for small values of proximity, the
better the small-set expansion of the interaction graph of the constraints, the
less sound the qLTC becomes. This phenomenon, which can be attributed to
monogamy of entanglement, stands in sharp contrast to the classical setting.
The complementary, more intuitive, result also holds: an upper bound on the
soundness when the code is defined on poor small-set expanders (a bound which
turns out to be far more difficult to show in the quantum case). Together we
arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any
graph, which does not hold in the classical case. Many open questions are
raised regarding what possible parameters are achievable for qLTCs. In the
appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and
point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with
related parameters, carries over to the sLTCs. This creates a first link
between qLTCs and quantum PCPs.Comment: Some of the results presented here appeared in an initial form in our
quant-ph submission arXiv:1301.3407. This is a much extended and improved
version. 30 pages, no figure
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries
A local tester for an error correcting code is a
tester that makes oracle queries to a given word and
decides to accept or reject the word . An optimal local tester is a local
tester that has the additional properties of completeness and optimal
soundness. By completeness, we mean that the tester must accept with
probability if . By optimal soundness, we mean that if the tester
accepts with probability at least (where is small),
then it must be the case that is -close to some codeword
in Hamming distance.
We show that Generalized Reed-Muller codes admit optimal testers with queries. Here, for a prime power , the Generalized Reed-Muller code, RM[n,q,d], consists of the
evaluations of all -variate degree polynomials over .
Previously, no tester achieving this query complexity was known, and the best
known testers due to Haramaty, Shpilka and Sudan(which is optimal) and due to
Ron-Zewi and Sudan(which was not known to be optimal) both required
queries. Our tester achieves query
complexity which is polynomially better than by a power of , which is
nearly the best query complexity possible for generalized Reed-Muller codes.
The tester we analyze is due to Ron-Zewi and Sudan, and we show that their
basic tester is in fact optimal. Our methods are more general and also allow us
to prove that a wide class of testers, which follow the form of the Ron-Zewi
and Sudan tester, are optimal. This result applies to testers for all
affine-invariant codes (which are not necessarily generalized Reed-Muller
codes).Comment: 42 pages, 8 page appendi
Algebraic Methods in Computational Complexity
From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 “Algebraic Methods in Computational Complexity “ was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
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