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We analyze metrics for how close an entire function of genus one is to having only real roots. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any poten- tial non-real zeros are forced to be farther and farther away from the critical line.
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Zero Deletion/Insertion Codes and Zero Error Capacity
In this paper the theory and design of codes capable of correcting t insertion/deletion of the symbol 0 in each and every bucket of zeros (i. e., zeros in between two consecutive ones) are studied. It is shown that this problem is related to the zero error capacity achieving codes in limited magnitude error channel. Close to optimal non-systematic code designs and the encoding/decoding algorithms are described.
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- Award ID(s):
- 2006571
- PAR ID:
- 10562485
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 978-1-6654-2159-1
- Page Range / eLocation ID:
- 986 to 991
- Subject(s) / Keyword(s):
- Codes , Systematics , Symbols , Process control , Turbo codes , Parity check codes , Encoding
- Format(s):
- Medium: X
- Location:
- Espoo, Finland
- Sponsoring Org:
- National Science Foundation
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This paper gives some theory and efficient design of binary block codes capable of controlling the deletions of the symbol “0” (referred to as 0-deletions) and/or the insertions of the symbol “0” (referred to as 0-insertions). This problem of controlling 0-deletions and/or 0-insertions (referred to as 0-errors) is shown to be equivalent to the efficient design of L1 metric asymmetric error control codes over the natural alphabet, IIN . In this way, it is shown that the t0 -insertion correcting codes are actually capable of controlling much more; namely, they can correct t0 -errors, detect (t+1)0 -errors and, simultaneously, detect all occurrences of only 0-deletions or only 0-insertions in every received word (briefly, they are t -Symmetric 0-Error Correcting/ (t+1) -Symmetric 0-Error Detecting/All Unidirectional 0-Error Detecting ( t -Sy0EC/ (t+1) -Sy0ED/AU0ED) codes). From the relations with the L1 distance error control codes, new improved bounds are given for the optimal t0 -error correcting codes. Optimal non-systematic code designs are given. Decoding can be efficiently performed by algebraic means using the Extended Euclidean Algorithm (EEA).more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ISIT50566.2022.9834523
