9 research outputs found
A short note on the joint entropy of n/2-wise independence
In this note, we prove a tight lower bound on the joint entropy of
unbiased Bernoulli random variables which are -wise independent. For
general -wise independence, we give new lower bounds by adapting Navon and
Samorodnitsky's Fourier proof of the `LP bound' on error correcting codes. This
counts as partial progress on a problem asked by Gavinsky and Pudl\'ak.Comment: 6 pages, some errors fixe
An entropy lower bound for non-malleable extractors
A (k, ε)-non-malleable extractor is a function nmExt : {0, 1} n × {0, 1} d → {0, 1} that takes two inputs, a weak source X ~ {0, 1} n of min-entropy k and an independent uniform seed s E {0, 1} d , and outputs a bit nmExt(X, s) that is ε-close to uniform, even given the seed s and the value nmExt(X, s') for an adversarially chosen seed s' ≠s. Dodis and Wichs (STOC 2009) showed the existence of (k, ε)-non-malleable extractors with seed length d = log(n - k - 1) + 2 log(1/ε) + 6 that support sources of min-entropy k > log(d) + 2 log(1/ε) + 8. We show that the foregoing bound is essentially tight, by proving that any (k, ε)-non-malleable extractor must satisfy the min-entropy bound k > log(d) + 2 log(1/ε) - log log(1/ε) - C for an absolute constant C. In particular, this implies that non-malleable extractors require min-entropy at least Ω(loglog(n)). This is in stark contrast to the existence of strong seeded extractors that support sources of min-entropy k = O(log(1/ε)). Our techniques strongly rely on coding theory. In particular, we reveal an inherent connection between non-malleable extractors and error correcting codes, by proving a new lemma which shows that any (k, ε)-non-malleable extractor with seed length d induces a code C ⊆ {0,1} 2k with relative distance 1/2 - 2ε and rate d-1/2k
Coding-Theoretic Methods for Sparse Recovery
We review connections between coding-theoretic objects and sparse learning
problems. In particular, we show how seemingly different combinatorial objects
such as error-correcting codes, combinatorial designs, spherical codes,
compressed sensing matrices and group testing designs can be obtained from one
another. The reductions enable one to translate upper and lower bounds on the
parameters attainable by one object to another. We survey some of the
well-known reductions in a unified presentation, and bring some existing gaps
to attention. New reductions are also introduced; in particular, we bring up
the notion of minimum "L-wise distance" of codes and show that this notion
closely captures the combinatorial structure of RIP-2 matrices. Moreover, we
show how this weaker variation of the minimum distance is related to
combinatorial list-decoding properties of codes.Comment: Added Lemma 34 in the first revision. Original version in Proceedings
of the Allerton Conference on Communication, Control and Computing, September
201
Estimating the Sizes of Binary Error-Correcting Constrained Codes
In this paper, we study binary constrained codes that are resilient to
bit-flip errors and erasures. In our first approach, we compute the sizes of
constrained subcodes of linear codes. Since there exist well-known linear codes
that achieve vanishing probabilities of error over the binary symmetric channel
(which causes bit-flip errors) and the binary erasure channel, constrained
subcodes of such linear codes are also resilient to random bit-flip errors and
erasures. We employ a simple identity from the Fourier analysis of Boolean
functions, which transforms the problem of counting constrained codewords of
linear codes to a question about the structure of the dual code. We illustrate
the utility of our method in providing explicit values or efficient algorithms
for our counting problem, by showing that the Fourier transform of the
indicator function of the constraint is computable, for different constraints.
Our second approach is to obtain good upper bounds, using an extension of
Delsarte's linear program (LP), on the largest sizes of constrained codes that
can correct a fixed number of combinatorial errors or erasures. We observe that
the numerical values of our LP-based upper bounds beat the generalized sphere
packing bounds of Fazeli, Vardy, and Yaakobi (2015).Comment: 51 pages, 2 figures, 9 tables, to be submitted to the IEEE Journal on
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