90 research outputs found
Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields
Reed-Muller codes are some of the oldest and most widely studied
error-correcting codes, of interest for both their algebraic structure as well
as their many algorithmic properties. A recent beautiful result of Saptharishi,
Shpilka and Volk showed that for binary Reed-Muller codes of length and
distance , one can correct random errors
in time (which is well beyond the worst-case error
tolerance of ).
In this paper, we consider the problem of `syndrome decoding' Reed-Muller
codes from random errors. More specifically, given the
-bit long syndrome vector of a codeword corrupted in
random coordinates, we would like to compute the
locations of the codeword corruptions. This problem turns out to be equivalent
to a basic question about computing tensor decomposition of random low-rank
tensors over finite fields.
Our main result is that syndrome decoding of Reed-Muller codes (and the
equivalent tensor decomposition problem) can be solved efficiently, i.e., in
time. We give two algorithms for this problem:
1. The first algorithm is a finite field variant of a classical algorithm for
tensor decomposition over real numbers due to Jennrich. This also gives an
alternate proof for the main result of Saptharishi et al.
2. The second algorithm is obtained by implementing the steps of the
Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in
sublinear-time. The main new ingredient is an algorithm for solving certain
kinds of systems of polynomial equations.Comment: 24 page
A Spectral Bound on Hypergraph Discrepancy
Let be a -regular hypergraph on vertices and edges.
Let be the incidence matrix of and let us denote
. We show that the
discrepancy of is . As a corollary, this
gives us that for every , the discrepancy of a random -regular hypergraph
with vertices and edges is almost surely as
grows. The proof also gives a polynomial time algorithm that takes a hypergraph
as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1811.01491, several changes to the presentatio
On the List Recoverability of Randomly Punctured Codes
We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes
Unbalanced Random Matching Markets with Partial Preferences
Properties of stable matchings in the popular random-matching-market model
have been studied for over 50 years. In a random matching market, each agent
has complete preferences drawn uniformly and independently at random. Wilson
(1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching
markets, the proposers are matched to their th choice on average. In
this paper, we consider markets where agents have partial (truncated)
preferences, that is, the proposers only rank their top partners. Despite
the long history of the problem, the following fundamental question remained
unanswered: \emph{what is the smallest value of that results in a perfect
stable matching with high probability?} In this paper, we answer this question
exactly -- we prove that a degree of is necessary and sufficient.
That is, we show that if then no stable matching is
perfect and if , then every stable matching is
perfect with high probability. This settles a recent conjecture by Kanoria, Min
and Qian (2021).
We generalize this threshold for unbalanced markets: we consider a matching
market with agents on the shorter side and agents on the
longer side. We show that for markets with , the sharp threshold
characterizing the existence of perfect stable matching occurs when is .
Finally, we extend the line of work studying the effect of imbalance on the
expected rank of the proposers (termed the ``stark effect of competition''). We
establish the regime in unbalanced markets that forces this stark effect to
take shape in markets with partial preferences
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