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Random Reed-Solomon Codes Achieve List Recovery Capacity with Optimal List Size


Core Concepts
Random Reed-Solomon codes are list recoverable up to capacity with optimal output list size, for any input list size.
Abstract
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.
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Deeper Inquiries

What is the minimum field size q required to achieve the list recovery parameters in Theorem 1.3, if the output list size L is allowed to be slightly larger, say polynomial in ℓ/ε

To achieve the list recovery parameters in Theorem 1.3 with the output list size L allowed to be slightly larger, say polynomial in ℓ/ε, we need the field size q to be at least ℓΘ(ℓ2/Rε3)·n2. This requirement ensures that the random RS codes can achieve the desired list recovery properties with a polynomial dependence on ℓ/ε for the output list size L.

Can the exponential dependence of q on ℓ be improved to a polynomial dependence, even if the output list size L is not optimal

It may be possible to improve the exponential dependence of the field size q on ℓ to a polynomial dependence even if the output list size L is not optimal. By relaxing the constraint on the output list size L to be slightly larger, such as polynomial in ℓ/ε, it opens up the possibility of reducing the exponential dependence on ℓ in the field size q. Further research and analysis would be needed to determine the exact conditions under which this improvement can be achieved.

How do the list recovery properties of random RS codes compare to those of other structured code families, such as folded RS codes or codes constructed using expanders, in terms of the trade-offs between rate, list recovery radius, input list size, output list size, and field size

Random RS codes exhibit favorable trade-offs between rate, list recovery radius, input list size, output list size, and field size compared to other structured code families like folded RS codes or codes constructed using expanders. Random RS codes, as shown in Theorem 1.3, can achieve list recovery capacity with optimal output list size and field size, providing a strong guarantee on their list recoverability. In contrast, folded RS codes and codes constructed using expanders may have different trade-offs and may not always achieve the same level of list recovery performance as random RS codes. Each code family has its strengths and weaknesses, and the choice of code depends on the specific requirements of the application.
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