Let's Have Both! Optimal List-Recoverability via Alphabet Permutation Codes

TL;DR

This paper introduces alphabet-permutation codes, a structured class of error-correcting codes that achieve near-optimal list-recoverability with significantly fewer random bits than fully random codes.

Contribution

The paper presents alphabet-permutation codes that match the list-recovery performance of random codes while requiring only polynomially many random bits, a novel structured approach.

Findings

Achieve almost sure list-recoverability with list size proportional to inverse gap to capacity.

Require only polynomially many random bits, unlike fully random codes.

Outperform linear codes in list size at the same rate.

Abstract

We introduce alphabet-permutation (AP) codes, a new family of error-correcting codes defined by iteratively applying random coordinate-wise permutations to a fixed initial word. A special case recovers random additive codes and random binary linear codes, where each permutation corresponds to an additive shift over a finite field. We show that when these permutations are drawn from a suitably ``mixing'' distribution, the resulting code is almost surely list-recoverable with list size proportional to the inverse of the gap to capacity. Compared to any linear code, our construction achieves exponentially smaller list sizes at the same rate. Previously, only fully random codes were known to attain such parameters, requiring exponentially many random bits and offering no structure. In contrast, AP codes are structured and require only polynomially many random bits -- providing the first such construction to match the list-recovery guarantees of random codes.