Combinatorial Limitations of Average-Radius List-Decoding
We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p (log(1/γ))) for the list size needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 - h(p) - γ [here p E (0, 1/2)...
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| Published in: | IEEE transactions on information theory Vol. 60; no. 10; pp. 5827 - 5842 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.10.2014
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p (log(1/γ))) for the list size needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 - h(p) - γ [here p E (0, 1/2) and γ > 0]. Our main result is that we prove that in any binary code C ⊆ (0, 1) n of rate 1 - h(p) - γ, there must exist a set l ⊂ C of p(1/√γ) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ω p (1/√γ) codewords with low average radius. The standard notion of list decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability. The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding as follows. First, we give a short simple proof, over all fixed alphabets, of the above-mentioned Ω p (log(1/γ)) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. Second, we show that one cannot improve the Ω p (log(1/γ)) lower bound via techniques based on identifying the zero-rate regime for list-decoding of constantweight codes [this is a typical approach for negative results in coding theory, including the Ω p (log(1/γ)) list-size lower bound]. On a positive note, our Ω p (1/√γ) lower bound for average radius list-decoding circumvents this barrier. Third, we exhibit a reverse connection between the existence of constant-weight and general codes for list-decoding, showing that the best possible list-size, as a function of the gap γ of the rate to the capacity limit, is the same up to constant factors for both constant-weight codes (with weight bounded away from p) and general codes. Fourth, we give simple second moment-based proofs that w.h.p. a list-size of Ω p (1/γ) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are Ω p (1/γ) for errors and expΩ p (log(1/γ)) for erasures. |
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| AbstractList | We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/...)) and lower bound (of ...(log(1/...))) for the list size needed to list decode up to error fraction p with rate ... away from capacity, i.e., 1 - h(p) - ... [here p E (0, 1/2) and ... > 0]. Our main result is that we prove that in any binary code C ... (0, 1)... of rate 1 - h(p) - ..., there must exist a set l ... C of p(1/...) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist ...(1/...) codewords with low average radius. The standard notion of list decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability. The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding as follows. First, we give a short simple proof, over all fixed alphabets, of the above-mentioned ...(log(1/...)) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. Second, we show that one cannot improve the ...(log(1/...)) lower bound via techniques based on identifying the zero-rate regime for list-decoding of constantweight codes [this is a typical approach for negative results in coding theory, including the ...(log(1/...)) list-size lower bound]. On a positive note, our ...(1/...) lower bound for average radius list-decoding circumvents this barrier. Third, we exhibit a reverse connection between the existenc- of constant-weight and general codes for list-decoding, showing that the best possible list-size, as a function of the gap ... of the rate to the capacity limit, is the same up to constant factors for both constant-weight codes (with weight bounded away from p) and general codes. Fourth, we give simple second moment-based proofs that w.h.p. a list-size of ...(1/...) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are ...(1/...) for errors and exp...(log(1/...)) for erasures. (ProQuest: ... denotes formulae/symbols omitted.) We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p (log(1/γ))) for the list size needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 - h(p) - γ [here p E (0, 1/2) and γ > 0]. Our main result is that we prove that in any binary code C ⊆ (0, 1) n of rate 1 - h(p) - γ, there must exist a set l ⊂ C of p(1/√γ) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ω p (1/√γ) codewords with low average radius. The standard notion of list decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability. The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding as follows. First, we give a short simple proof, over all fixed alphabets, of the above-mentioned Ω p (log(1/γ)) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. Second, we show that one cannot improve the Ω p (log(1/γ)) lower bound via techniques based on identifying the zero-rate regime for list-decoding of constantweight codes [this is a typical approach for negative results in coding theory, including the Ω p (log(1/γ)) list-size lower bound]. On a positive note, our Ω p (1/√γ) lower bound for average radius list-decoding circumvents this barrier. Third, we exhibit a reverse connection between the existence of constant-weight and general codes for list-decoding, showing that the best possible list-size, as a function of the gap γ of the rate to the capacity limit, is the same up to constant factors for both constant-weight codes (with weight bounded away from p) and general codes. Fourth, we give simple second moment-based proofs that w.h.p. a list-size of Ω p (1/γ) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are Ω p (1/γ) for errors and expΩ p (log(1/γ)) for erasures. |
| Author | Narayanan, Srivatsan Guruswami, Venkatesan |
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| Keywords | Combinatorial coding theory list error-correction linear codes random coding probabilistic method |
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| References | ref13 ref12 ref15 ref14 guruswami (ref9) 2013 ref10 ref2 cheraghchi (ref5) 2013 zyablov (ref17) 1981; 17 elias (ref6) 1957 ref8 ref7 ref4 ref3 blinovsky (ref1) 1986; 22 wozencraft (ref16) 1958; 48 guruswami (ref11) 2001 |
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| Snippet | We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p... We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/...)) and lower bound (of... |
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| SubjectTerms | Binary codes Coding theory Decoding Electronics Entropy Hamming distance Linear codes Points Upper bound Weight |
| Title | Combinatorial Limitations of Average-Radius List-Decoding |
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