Randomness-Efficient Constructions of Capacity-Achieving List-Decodable Codes

In this work, we consider the task of generating listdecodable codes over small (say, binary) alphabets using as little randomness as possible. Specifically, we hope to generate codes achieving what we term the Elias bound, which means that they are ($\rho, L$) -list-decodable with rate $R \geq 1-h(\rho)-O(1 / L)$. A long line of work shows that uniformly random linear codes (RLCs) achieve the Elias bound: hence, we know $O\left(n^{2}\right)$ random bits suffice. Prior works (Guruswami and Mosheiff, FOCS 2022; Putterman and Pyne, ITCS 2024) demonstrate that just $O(L n)$ random bits suffice, via puncturing of low-bias codes. These recent constructions are essentially combinatorial, and rely (directly or indirectly) on graph expansion. We provide two new constructions, which are algebraic. Compared to prior works, our constructions are considerably simpler and more direct. Furthermore, our codes are designed in such a way that their duals are also quite easy to analyze. Our first construction which can be seen as a generalization of the celebrated Wozencraft ensemble - achieves the Elias bound and consumes $L n$ random bits. Additionally, its dual code achieves the Gilbert-Varshamov bound with high probability, and both the primal and dual admit quasilinear-time encoding algorithms. The second construction consumes $2 L n$ random bits and yields a code where both it and its dual achieve the Elias bound. In all of the above cases - including the prior works achieving randomness complexity $O(L n)$ - the codes are designed to “approximate” RLCs. More precisely, for a given locality parameter $L$ we construct codes achieving the same $L$-local properties as RLCs. This allows one to appeal to known list-decodability results for RLCs and thereby conclude that the code approximating an RLC also achieves the Elias bound (with high probability). As a final contribution, we indicate that such a proof strategy is inherently unable to generate list-decodable codes of rate $R$ over $\mathbb{F}_{q}$ with less than $L(1-R) n \log _{2}(q)$ bits of randomness.