7 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
High rate locally-correctable and locally-testable codes with sub-polynomial query complexity
In this work, we construct the first locally-correctable codes (LCCs), and
locally-testable codes (LTCs) with constant rate, constant relative distance,
and sub-polynomial query complexity. Specifically, we show that there exist
binary LCCs and LTCs with block length , constant rate (which can even be
taken arbitrarily close to 1), constant relative distance, and query complexity
. Previously such codes were known to exist
only with query complexity (for constant ), and
there were several, quite different, constructions known.
Our codes are based on a general distance-amplification method of Alon and
Luby~\cite{AL96_codes}. We show that this method interacts well with local
correctors and testers, and obtain our main results by applying it to suitably
constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant
relative distance}.
Along the way, we also construct LCCs and LTCs over large alphabets, with the
same query complexity , which additionally have
the property of approaching the Singleton bound: they have almost the
best-possible relationship between their rate and distance. This has the
surprising consequence that asking for a large alphabet error-correcting code
to further be an LCC or LTC with query
complexity does not require any sacrifice in terms of rate and distance! Such a
result was previously not known for any query complexity.
Our results on LCCs also immediately give locally-decodable codes (LDCs) with
the same parameters
Relaxed Local Correctability from Local Testing
We cement the intuitive connection between relaxed local correctability and
local testing by presenting a concrete framework for building a relaxed locally
correctable code from any family of linear locally testable codes with
sufficiently high rate. When instantiated using the locally testable codes of
Dinur et al. (STOC 2022), this framework yields the first asymptotically good
relaxed locally correctable and decodable codes with polylogarithmic query
complexity, which finally closes the superpolynomial gap between query lower
and upper bounds. Our construction combines high-rate locally testable codes of
various sizes to produce a code that is locally testable at every scale: we can
gradually "zoom in" to any desired codeword index, and a local tester at each
step certifies that the next, smaller restriction of the input has low error.
Our codes asymptotically inherit the rate and distance of any locally
testable code used in the final step of the construction. Therefore, our
technique also yields nonexplicit relaxed locally correctable codes with
polylogarithmic query complexity that have rate and distance approaching the
Gilbert-Varshamov bound.Comment: 18 page
Locality via Partially Lifted Codes
In error-correcting codes, locality refers to several different ways of quantifying how easily a small amount of information can be recovered from encoded data. In this work, we study a notion of locality called the s-Disjoint-Repair-Group Property (s-DRGP). This notion can interpolate between two very different settings in coding theory: that of Locally Correctable Codes (LCCs) when s is large - a very strong guarantee - and Locally Recoverable Codes (LRCs) when s is small - a relatively weaker guarantee. This motivates the study of the s-DRGP for intermediate s, which is the focus of our paper. We construct codes in this parameter regime which have a higher rate than previously known codes. Our construction is based on a novel variant of the lifted codes of Guo, Kopparty and Sudan. Beyond the results on the s-DRGP, we hope that our construction is of independent interest, and will find uses elsewhere
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