7,811 research outputs found
Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices
In this paper we establish the connection between the Orthogonal Optical
Codes (OOC) and binary compressed sensing matrices. We also introduce
deterministic bipolar RIP fulfilling matrices of order
such that . The columns of these matrices are binary BCH code vectors where the
zeros are replaced by -1. Since the RIP is established by means of coherence,
the simple greedy algorithms such as Matching Pursuit are able to recover the
sparse solution from the noiseless samples. Due to the cyclic property of the
BCH codes, we show that the FFT algorithm can be employed in the reconstruction
methods to considerably reduce the computational complexity. In addition, we
combine the binary and bipolar matrices to form ternary sensing matrices
( elements) that satisfy the RIP condition.Comment: The paper is accepted for publication in IEEE Transaction on
Information Theor
Cyclotomic factors of the descent set polynomial
We introduce the notion of the descent set polynomial as an alternative way
of encoding the sizes of descent classes of permutations. Descent set
polynomials exhibit interesting factorization patterns. We explore the question
of when particular cyclotomic factors divide these polynomials. As an instance
we deduce that the proportion of odd entries in the descent set statistics in
the symmetric group S_n only depends on the number on 1's in the binary
expansion of n. We observe similar properties for the signed descent set
statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up
notatio
Efficient indexing of necklaces and irreducible polynomials over finite fields
We study the problem of indexing irreducible polynomials over finite fields,
and give the first efficient algorithm for this problem. Specifically, we show
the existence of poly(n, log q)-size circuits that compute a bijection between
{1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials
of degree n over a finite field F_q. This has applications in pseudorandomness,
and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP].
Our approach uses a connection between irreducible polynomials and necklaces
( equivalence classes of strings under cyclic rotation). Along the way, we give
the first efficient algorithm for indexing necklaces of a given length over a
given alphabet, which may be of independent interest
There are no Coincidences
This paper is inspired by a seqfan post by Jeremy Gardiner. The post listed
nine sequences with similar parities. In this paper I prove that the
similarities are not a coincidence but a mathematical fact.Comment: 9 page
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