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

We wish to generate list-decodable codes over small 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 $O(n^2)$ random bits suffice. Prior works demonstrate that just $O(Ln)$ random bits suffice, via puncturing of low-bias codes. These recent constructions are combinatorial. We provide two new constructions, which are algebraic. Compared to prior works, our constructions are 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 Wozencraft ensemble -- achieves the Elias bound and consumes $Ln$ random bits. Additionally, its dual code achieves the GV-bound with high probability, and both the primal and dual admit quasilinear-time encoding algorithms. The second construction consumes $2nL$ random bits and yields a code where both it and its dual achieve the Elias bound. As we discuss, properties of a dual code are often crucial for applications in cryptography. In all of the above cases -- including the prior works achieving randomness complexity $O(Ln)$ -- the codes are designed to "approximate" RLCs. Namely, 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. As a final contribution, we indicate that such a proof strategy is inherently unable to generate list-decodable codes of rate $R$ over $\mathbb Fq$ with less than $L(1-R)n\log2(q)$ bits of randomness.