Lower bound for the linear multiple packing of the binary Hamming space
Journal of Combinatorial Theory Series A
Asymptotic Combinatorial Coding Theory
Asymptotic Combinatorial Coding Theory
Expander-Based Constructions of Efficiently Decodable Codes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Code bounds for multiple packings over a nonbinary finite alphabet
Problems of Information Transmission
List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition (Lecture Notes in Computer Science)
On the convexity of one coding-theory function
Problems of Information Transmission
Combinatorial bounds for list decoding
IEEE Transactions on Information Theory
The existence of concatenated codes list-decodable up to the hamming bound
IEEE Transactions on Information Theory
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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We show that the list-decodability of random linear codes is as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0,1-1/q) and ε 0, we prove that with high probability a random linear code C in Fqn of rate (1-H_q(p)-ε) can be list decoded from a fraction p of errors with lists of size at most O(1/ε). This also answers a basic open question concerning the existence of highly list-decodable linear codes, showing that a list-size of O(1/ε) suffices to have rate within ε of the "list decoding capacity" 1-Hq(p). The best previously known list-size bound was qO(1/ε) (except in the q=2 case where a list-size bound of O(1/ε) was known). The main technical ingredient in our proof is a strong upper bound on the probability that l random vectors chosen from a Hamming ball centered at the origin have too many (more than Θ(l)) vectors from their linear span also belong to the ball.