Abstract
There has been a great deal of work establishing that random linear codes are as listdecodable as uniformly random codes, in the sense that a random linear binary code of rate 1  H(p)  epsilon is (p,O(1/epsilon))listdecodable with high probability. In this work, we show that such codes are (p, H(p)/epsilon + 2)listdecodable with high probability, for any p in (0, 1/2) and epsilon > 0. In addition to improving the constant in known listsize 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, HÃ¥stad, 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 listdecodable 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 rankmetric codes.
BibTeX  Entry
@InProceedings{li_et_al:LIPIcs:2018:9454,
author = {Ray Li and Mary Wootters},
title = {{Improved ListDecodability of Random Linear Binary Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {50:150:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770859},
ISSN = {18688969},
year = {2018},
volume = {116},
editor = {Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9454},
URN = {urn:nbn:de:0030drops94547},
doi = {10.4230/LIPIcs.APPROXRANDOM.2018.50},
annote = {Keywords: Listdecoding, Random linear codes, Rankmetric codes}
}
Keywords: 

Listdecoding, Random linear codes, Rankmetric codes 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) 
Issue Date: 

2018 
Date of publication: 

13.08.2018 