228 research outputs found
It'll probably work out: improved list-decoding through random operations
In this work, we introduce a framework to study the effect of random
operations on the combinatorial list-decodability of a code. The operations we
consider correspond to row and column operations on the matrix obtained from
the code by stacking the codewords together as columns. This captures many
natural transformations on codes, such as puncturing, folding, and taking
subcodes; we show that many such operations can improve the list-decoding
properties of a code. There are two main points to this. First, our goal is to
advance our (combinatorial) understanding of list-decodability, by
understanding what structure (or lack thereof) is necessary to obtain it.
Second, we use our more general results to obtain a few interesting corollaries
for list decoding:
(1) We show the existence of binary codes that are combinatorially
list-decodable from fraction of errors with optimal rate
that can be encoded in linear time.
(2) We show that any code with relative distance, when randomly
folded, is combinatorially list-decodable fraction of errors with
high probability. This formalizes the intuition for why the folding operation
has been successful in obtaining codes with optimal list decoding parameters;
previously, all arguments used algebraic methods and worked only with specific
codes.
(3) We show that any code which is list-decodable with suboptimal list sizes
has many subcodes which have near-optimal list sizes, while retaining the error
correcting capabilities of the original code. This generalizes recent results
where subspace evasive sets have been used to reduce list sizes of codes that
achieve list decoding capacity
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
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}
On the List-Decodability of Random Linear Codes
For every fixed finite field \F_q, and , we
prove that with high probability a random subspace of \F_q^n of dimension
has the property that every Hamming ball of radius
has at most codewords.
This answers a basic open question concerning the list-decodability of linear
codes, showing that a list size of suffices to have rate within
of the "capacity" . Our result matches up to constant
factors the list-size achieved by general random codes, and gives an
exponential improvement over the best previously known list-size bound of
.
The main technical ingredient in our proof is a strong upper bound on the
probability that random vectors chosen from a Hamming ball centered at
the origin have too many (more than ) vectors from their linear
span also belong to the ball.Comment: 15 page
Efficiently decodable non-adaptive group testing
We consider the following "efficiently decodable" non-adaptive
group testing problem. There is an unknown string
x 2 f0; 1gn [x is an element of set {0,1} superscript n] with at most d ones in it. We are allowed to test
any subset S [n] [S subset [n] ]of the indices. The answer to the test
tells whether xi = 0 [x subscript i = 0] for all i 2 S [i is an element of S] or not. The objective
is to design as few tests as possible (say, t tests) such that
x can be identifi ed as fast as possible (say, poly(t)-time).
Efficiently decodable non-adaptive group testing has applications
in many areas, including data stream algorithms and
data forensics.
A non-adaptive group testing strategy can be represented
by a t x n matrix, which is the stacking of all the
characteristic vectors of the tests. It is well-known that if
this matrix is d-disjunct, then any test outcome corresponds
uniquely to an unknown input string. Furthermore, we know
how to construct d-disjunct matrices with t = O(d2 [d superscript 2] log n)
efficiently. However, these matrices so far only allow for a
"decoding" time of O(nt), which can be exponentially larger
than poly(t) for relatively small values of d.
This paper presents a randomness efficient construction
of d-disjunct matrices with t = O(d2 [d superscript 2] log n) that can be decoded
in time poly(d) [function composed of] t log2 t + O(t2) [t log superscript 2 t and O (t superscript 2)]. To the best of our
knowledge, this is the first result that achieves an efficient decoding
time and matches the best known O(d2 log n) [O (d superscript 2 log n)] bound
on the number of tests. We also derandomize the construction,
which results in a polynomial time deterministic construction
of such matrices when d = O(log n= log log n).
A crucial building block in our construction is the
notion of (d,l)-list disjunct matrices, which represent the
more general "list group testing" problem whose goal is to
output less than d + l positions in x, including all the (at
most d) positions that have a one in them. List disjunct
matrices turn out to be interesting objects in their own right
and were also considered independently by [Cheraghchi,
FCT 2009]. We present connections between list disjunct
matrices, expanders, dispersers and disjunct matrices. List
disjunct matrices have applications in constructing (d,l)-
sparsity separator structures [Ganguly, ISAAC 2008] and in
constructing tolerant testers for Reed-Solomon codes in the
data stream model.
1 IntroductionDavid & Lucile Packard FoundationCenter for Massive Data Algorithmics (MADALGO)National Science Foundation (U.S.) (Grant CCF-0728645)National Science Foundation (U.S.) (Grant CCF-0347565)National Science Foundation (U.S.) (CAREER Award CCF-0844796
Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction
A method for concatenating quantum error-correcting codes is presented. The
method is applicable to a wide class of quantum error-correcting codes known as
Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate
in the Shannon theoretic sense and that are decodable in polynomial time are
presented. The rate is the highest among those known to be achievable by CSS
codes. Moreover, the best known lower bound on the greatest minimum distance of
codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of
the AE of the journal, the present version has become a combination of
(thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195.
Problem formulations of polynomial complexity are strictly followed. An
erroneous instance of a lower bound on minimum distance was remove
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