When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?
Dean Doron, Jonathan Mosheiff, Mary Wootters
TL;DR
The paper investigates when low-rate concatenated codes can approach the Gilbert–Varshamov bound using a single random inner binary code. It develops a moment-based analysis that expresses codeword bias X_m via the dual outer code and Fourier methods, and shows that with high probability a random (or suitably smooth) outer code C_out concatenated with a single random inner code C_in yields a code C_out ∘ C_in with rate R = Ω(ε^2) and distance δ = 1/2 − O(ε). It introduces two sufficient outer-code conditions—one soft-decoding-like on C_out^⊥ and one based on smooth min-entropy of C_out—that guarantee GV-bound proximity for the concatenated code, providing a framework toward explicit outer-code constructions. The results unify and extend prior work on explicit concatenations, random-outer constructions, and low-randomness GV-bound results, offering a path to derandomizing GV-bound constructions via structured outer codes and single-inner-code randomness with potential practical decoding implications.
Abstract
The Gilbert--Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate $ε^2$ has relative distance at least $\frac{1}{2} - O(ε)$ with high probability. However, it is a major challenge to construct explicit codes with similar parameters. One hope to derandomize the Gilbert--Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code ${C}_\mathrm{out}$ over a large alphabet, and concatenate that with a small binary random linear code ${C}_\mathrm{in}$. It is known that when we use independent small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code ${C}_\mathrm{in}$ can lie on the GV bound; and if so what conditions on ${C}_\mathrm{out}$ are sufficient for this. We show that first, there do exist linear outer codes ${C}_\mathrm{out}$ that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for ${C}_\mathrm{out}$, so that if ${C}_\mathrm{out}$ satisfies these, ${C}_\mathrm{out}\circ {C}_\mathrm{in}$ will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes ${C}_\mathrm{out}$.