5,422 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
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The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings
We review the concept of the number wall as an alternative to the traditional
linear complexity profile (LCP), and sketch the relationship to other topics
such as linear feedback shift-register (LFSR) and context-free Lindenmayer
(D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is
introduced having deficiency 2 modulo 3, and this property verified via the
re-interpretation of the number wall as an aperiodic plane tiling
Proofs of two conjectures on ternary weakly regular bent functions
We study ternary monomial functions of the form f(x)=\Tr_n(ax^d), where
x\in \Ff_{3^n} and \Tr_n: \Ff_{3^n}\to \Ff_3 is the absolute trace
function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss
sums, and certain ternary weight inequalities, we show that certain ternary
monomial functions arising from \cite{hk1} are weakly regular bent, settling a
conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the
Coulter-Matthews bent functions are weakly regular.Comment: 20 page
Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories
Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation
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