Combinatorial Limitations of Average-Radius List-Decoding

Abstract

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, 1 2 ) and γ > 0). Our main result is the following:• We prove that in any binary code C ⊆ {0,codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ω p (1/ √ γ) codewords with low "average radius."The standard notion of list-decoding corresponds to working with the maximum 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 listdecodability.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 (this is a typical approach for negative results in coding theory, including the Ω p (log(1/γ)) list-size lower bound). On a positive note, our Ω p (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 (with weight 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 listdecoding 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.

List Decoding Random Euclidean Codes and Infinite Constellations

“…Indeed, it is wellknown and easy to show that the list size of a random code (of which each codeword is sampled uniformly and independently) of rate 1´H q ppq´δ against a power-p adversary is at most 1{δ with high probability (whp). It also turns out [9], [10] that 1{δ is the correct scaling for the list size of a random code. Namely, there is an essentially matching 1 lower bound 1{δ via second moment method.…”

“…Namely, if we sample a linear code uniformly at random, its list size is Ωp1{δq whp. Nonetheless, in general, the best lower bound on list size for any code is still Ωplogp1{δqq [18], [19], [20], [21], [9] -there is an exponential gap between the best upper and lower bounds even over F 2 . Closing this gap is also a long standing open problem.…”

“…The achievability part again follows from a random coding argument and the scaling Θp1{δq of the list size of a random code is tight whp via a second moment computation [9]. For general codes, it can be shown that the list size is at least Ωplogp1{δqq using an erasure version of the punctured Plotkin-type bound (see, e.g., [3], Theorem 10.8).…”

“…On the other hand, if we restrict our attention to linear codes, the situation seems a little worse. The list size of a random linear code turns out to be 2 Θp1{δq whp (see, e.g., [3], Theorem 10.6 for an upper bound and [9] for a matching lower bound). Intuitively, for a linear code C, the list x P C : x| T " y| T ( corresponding to a received word y P pF q Y t?uq n with p1´pqn unerased locations in T Ď rns forms an affine subspace.…”

“…1) Careful readers might have already noted that a missing piece in this work is a list size lower bound for random Construction-A lattice codes. We had trouble replicating the arguments in [9]. We leave it as an open question to get a polynomial-in-1{δ list size lower bound.…”