3,243 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
Is Unemployment Good for the Environment?
Environmental quality is a public good, potentially impacted by everybody. Individual level pro-environmental behavior affects environmental quality in the aggregate. Therefore, it is important to understand what causes individual’s pro-environmental behaviors to change. We quantify the causal effect of one determinant, unemployment, using an EU-27 population representative Eurobarometer survey. Drawing on results from the theory of the private provision of public goods, and recognizing that unemployment decreases income and the opportunity cost of time, we formulate testable predictions that unemployment will decrease the extent of pro-environmental behaviors that require monetary contributions and increase the extent of pro-environmental behaviors that mainly require time/effort. Instrumental variables regressions provide empirical evidence to support these hypotheses. Changes in the unemployment rate within a sub-national region provide the exogenous variation needed to identify the causal effect. Several supplemental questions on the survey provide evidence that environmental issues lose saliency and economic issues gain saliency when one becomes unemployed, suggesting that interested parties may wish to emphasize cost savings of pro-environmental behavior rather than environmental benefits during times of increased unemployment
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
Interactive Oracle Proofs with Constant Rate and Query Complexity
We study interactive oracle proofs (IOPs) [BCS16,RRR16], which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs or IPs alone. Our main results are:
1. Circuit satisfiability has 3-round IOPs with linear proof length (counted in bits) and constant query complexity.
2. Reed-Solomon codes have 2-round IOPs of proximity with linear proof length and constant query complexity.
3. Tensor product codes have 1-round IOPs of proximity with sublinear proof length and constant query complexity.
For all the above, known PCP constructions give quasilinear proof length and constant query complexity [BS08,Din07]. Also, for circuit satisfiability, [BKKMS13] obtain PCPs with linear proof length but sublinear (and super-constant) query complexity. As in [BKKMS13], we rely on algebraic-geometry codes to obtain our first result; but, unlike that work, our use of such codes is much "lighter" because we do not rely on any automorphisms of the code.
We obtain our results by proving and combining "IOP-analogues" of tools underlying numerous IPs and PCPs:
* Interactive proof composition. Proof composition [AS98] is used to reduce the query complexity of PCP verifiers, at the cost of increasing proof length by an additive factor that is exponential in the verifier\u27s randomness complexity. We prove a composition theorem for IOPs where this additive factor is linear.
* Sublinear sumcheck. The sumcheck protocol [LFKN92] is an IP that enables the verifier to check the sum of values of a low-degree multi-variate polynomial on an exponentially-large hypercube, but the verifier\u27s running time depends linearly on the bound on individual degrees. We prove a sumcheck protocol for IOPs where this dependence is sublinear (e.g., polylogarithmic).
Our work demonstrates that even constant-round IOPs are more efficient than known PCPs and IPs
Robust Quantum Entanglement at (Nearly) Room Temperature
We formulate a mixed-state analog of the NLTS conjecture [FH14] by asking
whether there exist topologically-ordered systems for which the thermal Gibbs
state for constant temperature is globally-entangled in the sense that it
cannot even be approximated by shallow quantum circuits. We then prove this
conjecture holds for nearly optimal parameters: when the "inverse temperature"
is almost a constant (temperature decays as 1/loglog(n))) and the Hamiltonian
is nearly local (log(n)-local). The construction and proof combine quantum
codes that arise from high-dimensional manifolds [Has17, LLZ19], the
local-decoding approach to quantum codes [LTZ15, FGL18] and quantum
locally-testable codes [AE15].Comment: Strengthened main theorem, small modifications to the proof, revised
introductio
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