113 research outputs found
Revisiting the Sanders-Freiman-Ruzsa Theorem in and its Application to Non-malleable Codes
Non-malleable codes (NMCs) protect sensitive data against degrees of
corruption that prohibit error detection, ensuring instead that a corrupted
codeword decodes correctly or to something that bears little relation to the
original message. The split-state model, in which codewords consist of two
blocks, considers adversaries who tamper with either block arbitrarily but
independently of the other. The simplest construction in this model, due to
Aggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bit
messages to -bit codewords. It is conjectured, however, that the
construction allows linear-length codewords. Towards resolving this conjecture,
we show that the construction allows for code-length . This is achieved
by analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for general
Abelian groups. Closely following the excellent exposition of this result for
the group by Lovett, we expose its dependence on for the
group , where is a prime
Gaussian width bounds with applications to arithmetic progressions in random settings
Motivated by problems on random differences in Szemer\'{e}di's theorem and on
large deviations for arithmetic progressions in random sets, we prove upper
bounds on the Gaussian width of point sets that are formed by the image of the
-dimensional Boolean hypercube under a mapping
, where each coordinate is a constant-degree
multilinear polynomial with 0-1 coefficients. We show the following
applications of our bounds. Let be the random
subset of containing each element independently with
probability .
A set is -intersective if
any dense subset of contains a proper -term
arithmetic progression with common difference in . Our main result implies
that is -intersective with probability provided for . This gives a polynomial improvement for all
of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and
reproves more directly the same improvement shown recently by the authors and
Dvir.
Let be the number of -term arithmetic progressions in
and consider the large deviation rate
. We give quadratic
improvements of the best-known range of for which a highly precise estimate
of due to Bhattacharya, Ganguly, Shao and Zhao is valid for
all odd .
We also discuss connections with error correcting codes (locally decodable
codes) and the Banach-space notion of type for injective tensor products of
-spaces.Comment: 18 pages, some typos fixe
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost
of a factor of two in the total communication, Alice could just as well have
given the two possible replies without listening and have Bob select which
applies to him. Motivated by a conjecture stating that this form of "round
elimination" is impossible in exact quantum communication complexity, we study
the orthogonal rank and a symmetric variant thereof for a certain family of
Cayley graphs. The orthogonal rank of a graph is the smallest number for
which one can label each vertex with a nonzero -dimensional complex vector
such that adjacent vertices receive orthogonal vectors.
We show an exp lower bound on the orthogonal rank of the graph on
in which two strings are adjacent if they have Hamming distance at
least . In combination with previous work, this implies an affirmative
answer to the above conjecture.Comment: 13 page
Failure of the trilinear operator space Grothendieck theorem
We give a counterexample to a trilinear version of the operator space
Grothendieck theorem. In particular, we show that for trilinear forms on
, the ratio of the symmetrized completely bounded norm and the
jointly completely bounded norm is in general unbounded, answering a question
of Pisier. The proof is based on a non-commutative version of the generalized
von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi
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
Locally Decodable Quantum Codes
We study a quantum analogue of locally decodable error-correcting codes. A
q-query locally decodable quantum code encodes n classical bits in an m-qubit
state, in such a way that each of the encoded bits can be recovered with high
probability by a measurement on at most q qubits of the quantum code, even if a
constant fraction of its qubits have been corrupted adversarially. We show that
such a quantum code can be transformed into a classical q-query locally
decodable code of the same length that can be decoded well on average (albeit
with smaller success probability and noise-tolerance). This shows, roughly
speaking, that q-query quantum codes are not significantly better than q-query
classical codes, at least for constant or small q.Comment: 15 pages, LaTe
Subspaces of tensors with high analytic rank
It is shown that for any subspace V⊆Fn×⋯×np of d-tensors, if dim(V)≥tnd−1, then there is subspace W⊆V of dimension at least t/(dr)−1 whose nonzero elements all have analytic rank Ωd,p(r). As an application, we generalize a result of Altman on Szemerédi's theorem with random differences
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
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