When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound? (2405.08584v3)

Abstract: The Gilbert--Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate $\epsilon^2$ has relative distance at least $\frac{1}{2} - O(\epsilon)$ 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 ${C}\mathrm{out}$.