When Do Low-Rate Concatenated Codes Approach The Gilbert–Varshamov Bound?

The Gilbert–Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate ε² has relative distance at least ¹₂ − 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 Cout over a large alphabet, and concatenate that with a small binary random linear code Cin. 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 Cin can lie on the GV bound; and if so what conditions on Cout are sufficient for this. We show that first, there do exist linear outer codes Cout that are “good” for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for Cout, so that if Cout satisfies these, Cout ◦ Cin will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes Cout.