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Limits to List Decoding of Random Codes

机译：随机码列表解码的限制

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摘要

It has been known since [Zyablov and Pinsker, 1982] that a random $q$-ary code of rate $1-H_{q}(rho )- varepsilon $ (where $00$ is small enough and $H_{q}(cdot )$ is the $q$-ary entropy function) with high probability is a $(rho ,1/ varepsilon )$-list decodable code (that is, every Hamming ball of radius at most $rho n$ has at most $1/ varepsilon $ codewords in it). In this paper, the “converse” result is proven. In particular, it is proven that for every $0 $rho geqslant 1-1/q-O(sqrt { varepsilon })$ for small enough $ varepsilon >0$ [Blinovsky, 1986, 2005, 2008; Guruswami and Vadhan, 2005]. A lower bound is known for all constant $0< rho < 1-1/q$ independent of $ varepsilon $, though the lower bound is asymptotically weaker than our bound [Blinovsky, 1986, 2005, 2008]. These results, however, are not subsumed by ours as these other results hold for arbitrary codes of rate $1-H_{q}(rho )- varepsilon $.
机译：自[Zyablov and Pinsker，1982]以来，已经知道速率为$ 1-H_ {q}（rho）-varepsilon $（其中$ 00 $足够小而$ H_ {q}（cdot ）$是$ q $进制的熵函数）高概率是$（rho，1 / varepsilon）$列表可解码的代码（即，半径为$ rho n $的每个汉明球最多具有$ 1 / varepsilon $ codewords）。本文证明了“相反”的结果。特别地，已经证明，对于足够小的$ varepsilon> 0 $，对于每$ 0 $ r geqslant 1-1 / q-O（sqrt {varepsilon}）$ [Blinovsky，1986，2005，2008; Guruswami和Vadhan，2005年]。与常数varepsilon $无关，所有常数$ 0

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