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# Improved List-Decodability of Random Linear Binary Codes

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## Cite As

Ray Li and Mary Wootters. Improved List-Decodability of Random Linear Binary Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.50

## Abstract

There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate 1 - H(p) - epsilon is (p,O(1/epsilon))-list-decodable with high probability. In this work, we show that such codes are (p, H(p)/epsilon + 2)-list-decodable with high probability, for any p in (0, 1/2) and epsilon > 0. In addition to improving the constant in known list-size bounds, our argument - which is quite simple - works simultaneously for all values of p, while previous works obtaining L = O(1/epsilon) patched together different arguments to cover different parameter regimes. Our approach is to strengthen an existential argument of (Guruswami, Håstad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. To complement our upper bound for random linear binary codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain a tight lower bound of 1/epsilon on the list size of uniformly random binary codes; this implies that random linear binary codes are in fact more list-decodable than uniformly random binary codes, in the sense that the list sizes are strictly smaller. To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear binary codes and (b) to prove a similar result for random linear rank-metric codes.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Error-correcting codes
##### Keywords
• List-decoding
• Random linear codes
• Rank-metric codes

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## References

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