3 research outputs found
Relaxed locally correctable codes with nearly-linear block length and constant query complexity
Locally correctable codes (LCCs) are codes C: Σk → Σn which admit local algorithms that can correct any individual symbol of a corrupted codeword via a minuscule number of queries. One of the central problems in algorithmic coding theory is to construct O(1)-query LCC with minimal block length. Alas, state-of-the-art of such codes requires exponential block length to admit O(1)-query algorithms for local correction, despite much attention during the last two decades.
This lack of progress prompted the study of relaxed LCCs, which allow the correction algorithm to abort (but not err) on small fraction of the locations. This relaxation turned out to allow constant-query correction algorithms for codes with polynomial block length. Specifically, prior work showed that there exist O(1)-query relaxed LCCs that achieve nearly-quartic block length n = k4+α, for an arbitrarily small constant α > 0.
We construct an O(1)-query relaxed LCC with nearly-linear block length n = k1+α, for an arbitrarily small constant α > 0. This significantly narrows the gap between the lower bound which states that there are no O(1)-query relaxed LCCs with block length n = k1+o(1). In particular, this resolves an open problem raised by Gur, Ramnarayan, and Rothblum (ITCS 2018)
On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors
Locally Decodable Codes (LDCs) are error-correcting codes
with super-fast decoding algorithms. They are
important mathematical objects in many areas of theoretical computer science,
yet the best constructions so far have codeword length that is
super-polynomial in , for codes with constant query complexity and constant
alphabet size. In a very surprising result, Ben-Sasson et al. showed how to
construct a relaxed version of LDCs (RLDCs) with constant query complexity and
almost linear codeword length over the binary alphabet, and used them to obtain
significantly-improved constructions of Probabilistically Checkable Proofs. In
this work, we study RLDCs in the standard Hamming-error setting, and introduce
their variants in the insertion and deletion (Insdel) error setting. Insdel
LDCs were first studied by Ostrovsky and Paskin-Cherniavsky, and are further
motivated by recent advances in DNA random access bio-technologies, in which
the goal is to retrieve individual files from a DNA storage database. Our first
result is an exponential lower bound on the length of Hamming RLDCs making 2
queries, over the binary alphabet. This answers a question explicitly raised by
Gur and Lachish. Our result exhibits a "phase-transition"-type behavior on the
codeword length for constant-query Hamming RLDCs. We further define two
variants of RLDCs in the Insdel-error setting, a weak and a strong version. On
the one hand, we construct weak Insdel RLDCs with with parameters matching
those of the Hamming variants. On the other hand, we prove exponential lower
bounds for strong Insdel RLDCs. These results demonstrate that, while these
variants are equivalent in the Hamming setting, they are significantly
different in the insdel setting. Our results also prove a strict separation
between Hamming RLDCs and Insdel RLDCs
Testing Distributions of Huge Objects
We initiate a study of a new model of property testing that is a hybrid of
testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but
these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these
objects (resp., queries to the strings) as well as for the distance between
objects (resp., strings), and the distance between distributions is defined as
the earth mover's distance with respect to the relative Hamming distance
between strings.
We study the query complexity of testing in this new model, focusing on three
directions. First, we try to relate the query complexity of testing properties
in the new model to the sample complexity of testing these properties in the
standard distribution testing model. Second, we consider the complexity of
testing properties that arise naturally in the new model (e.g., distributions
that capture random variations of fixed strings). Third, we consider the
complexity of testing properties that were extensively studied in the standard
distribution testing model: Two such cases are uniform distributions and pairs
of identical distributions