Date of Original Version
82013
Type
Conference Proceeding
Rights Management
The final publication is available at Springer via http://dx.doi.org/10.1007/9783642403286_41
Abstract or Description
We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p (log(1/γ))) for the listsize needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 − h(p) − γ(here p∈(0,12) and γ > 0). Our main result is the following:
We prove that in any binary code C ⊆ {0, 1} n of rate 1 − h(p) − γ, there must exist a set L⊂C of Ωp(1/γ√) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ωp(1/γ√) codewords with low “average radius.” The standard notion of listdecoding corresponds to working with themaximum distance of a collection of codewords from a center instead of average distance. The averageradius form is in itself quite natural; for instance, the classical Johnson bound in fact implies averageradius listdecodability.
The remaining results concern the standard notion of listdecoding, and help clarify the current state of affairs regarding combinatorial bounds for listdecoding:

We give a short simple proof, over all fixed alphabets, of the abovementioned Ω p(log(1/γ)) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky.

We show that one cannot improve the Ω p (log(1/γ)) lower bound via techniques based on identifying the zerorate regime for listdecoding of constantweight codes. On a positive note, our Ωp(1/γ√) lower bound for averageradius listdecoding circumvents this barrier.

We exhibit a “reverse connection” between the existence of constantweight and general codes for listdecoding, showing that the best possible listsize, as a function of the gap γ of the rate to the capacity limit, is the same up to constant factors for both constantweight codes (whose weight is bounded away from p) and general codes.

We give simple second moment based proofs that w.h.p. a listsize of Ω p (1/γ) is needed for listdecoding random codes from errors as well as erasures. For random linear codes, the corresponding listsize bounds are Ω p (1/γ) for errors and exp(Ω p (1/γ)) for erasures.
DOI
10.1007/9783642403286_41
Published In
Lecture Notes in Computer Science, 8096, 591606.