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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| Vydáno v: | IEEE transactions on information theory Ročník 60; číslo 10; s. 5827 - 5842 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
IEEE
01.10.2014
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Témata: | |
| ISSN: | 0018-9448, 1557-9654 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0018-9448 1557-9654 |
| DOI: | 10.1109/TIT.2014.2343224 |