Combinatorial Limitations of Average-radius List Decoding

We study certain combinatorial aspects of list-decoding, 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 list-size 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 list-decoding corresponds to working with themaximum distance of a collection of codewords from a center instead of average distance. The average-radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability.

The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding:

• We give a short simple proof, over all fixed alphabets, of the above-mentioned 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 zero-rate regime for list-decoding of constant-weight codes. On a positive note, our Ωp(1/γ√) lower bound for average-radius list-decoding circumvents this barrier.

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

• We give simple second moment based proofs that w.h.p. a list-size of Ω p (1/γ) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are Ω p (1/γ) for errors and exp(Ω p (1/γ)) for erasures.