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

In this work, we consider the task of generating list-decodable 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 (ρ,L)-list-decodable with rate R ≥ 1-h(ρ)-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 (Guruswami and Mosheiff, FOCS 2022; Putterman and Pyne, arXiv 2023) demonstrate that just O_L(n) random bits suffice. We provide two new constructions achieving the Elias bound consuming only O(nL) random bits. 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. Of particular note, we construct with O(nL) random bits a code where both the code and its dual achieve the Elias bound! As we discuss, properties of a dual code are often crucial in applications of error-correcting codes in cryptography. 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 with o(nL) bits of randomness.