Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes
نویسندگان
چکیده
منابع مشابه
Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes
We prove that a random linear code over Fq, with probability arbitrarily close to 1, is list decodable at radius 1− 1/q− with list size L = O(1/ ) and rate R = Ωq( 2/(log(1/ ))). Up to the polylogarithmic factor in 1/ and constant factors depending on q, this matches the lower bound L = Ωq(1/ ) for the list size and upper bound R = Oq( ) for the rate. Previously only existence (and not abundanc...
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Ñ . These results are shown by choosing the code at random. However, as we saw in Chapter 4 the explicit constructions of codes over finite alphabets are nowhere close to achieving list-decoding capacity. The linear codes in Chapter 4 are based on code concatenation. A natural question to ask is whether linear codes based on code concatenation can get us to list-decoding capacity for fixed al...
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The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an Fq-linear rank-metric code over F m×n q of rate R = (1 − ρ)(1 − n m ρ) − ε is shown to be (with high probability) list-decodable up to fractional radius ρ ∈ (0, 1) with lists of size at most Cρ,q ε , where Cρ,q is a constant depending only on ρ and q. This matches the bo...
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A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ε if it preserves the l2 norm of all k-sparse vectors up to a factor of 1± ε. We prove that a matrix A obtained by randomly sampling q = O(k · log k · logN) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson an...
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In this paper we look at isometry properties of random matrices. During the last decade these properties gained a lot attention in a field called compressed sensing in first place due to their initial use in [7, 8]. Namely, in [7, 8] these quantities were used as a critical tool in providing a rigorous analysis of l1 optimization’s ability to solve an under-determined system of linear equations...
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ژورنال
عنوان ژورنال: SIAM Journal on Computing
سال: 2013
ISSN: 0097-5397,1095-7111
DOI: 10.1137/120896773