核心概念
Random Reed-Solomon codes are list recoverable up to capacity with optimal output list size, for any input list size.
要約
The paper proves that random Reed-Solomon (RS) codes are list recoverable up to capacity with optimal output list size, for any input list size. Specifically:
For any positive integers n, ℓ, any small enough ε > 0, and any rate R ∈ (0, 1-ε), the authors show that a random RS code RS(α1, ..., αn; Rn) is (1-R-ε, ℓ, L=O(ℓ/ε)) list recoverable with high probability, over a finite field Fq with q ≥ ℓᶿ(ℓ²/Rε³) · n².
This result improves upon previous work in several aspects:
The rate breaks the 1/ℓ barrier of the Johnson bound, while having optimal dependence on the gap to capacity ε.
The output list size L=O(ℓ/ε) matches the bound for random unstructured codes, beating the previous best bound for random linear codes.
The field size q is exponential in ℓ, which is necessary for the optimal list size, but the authors conjecture that a polynomial dependence on ℓ may be possible with a slightly worse list size.
The proof builds upon and extends the recent techniques for list decoding of random RS codes, introducing the notion of an "extended reduced intersection matrix" to handle the list recovery setting.