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SIMULA common base language by Ole-Johan Dahl

By Ole-Johan Dahl

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The drawback of the approach above is that one can only prove the required tight lower bound on the list size for ρ < 1/2(1 − 1/q). To push up ρ close to 1 − 1/q, we will need the following idea. Instead of carving the space of received words into (exponentially many) Hamming balls of relative radius 2ρ, we will carve the space into exponentially many disjoint clusters with the following properties. Every vector in a cluster is within a relative Hamming distance of ρ from the cluster center. Further, the size of every cluster is very close to the volume of a Hamming ball of relative radius ρ.

APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 318–329. Springer, Heidelberg (2005) 7. : Error Detecting and Error Correcting Codes. Bell System Technical Journal 29, 147–160 (1950) 8. : The Theory of Error-Correcting Codes. Elsevier/North-Holland, Amsterdam (1981) 9. : List Decoding and Property Testing of Error Correcting Codes. PhD thesis, University of Washington (2007) 10. : Limits to list decoding random codes. Electronic Colloquium on Computational Complexity (ECCC) 16(013) (2009) 11.

By definition, we have E Z 2 = m1 ,m2 ∈M Pr[Ym1 = 1 and Ym2 = 1]. By abuse of notation, for every m1 , m2 ∈ M, define m1 ∩ m2 34 A. Rudra to be the set of vectors from [q]k that occur in both the tuples m1 and m2 . Similarly, m1 ∪ m2 will denote the set of vectors from [q]k that occur in m1 or m2 . With this notation in place, let us rewrite the summation above: L 2 E Z = Pr[Ym1 = 1 Ym2 = 1] + i=1 m1 ,m2 ∈M Pr[Ym1 = 1 Ym2 = 1] m1 ,m2 ∈M m1 ∩m2=∅ |m1 ∩m2 |=i (11) We will bound the two summations in (11) using the following observation.

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