Repairing Reed-Solomon Codes via Subspace Polynomials
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We propose new repair schemes for Reed-Solomon codes that use subspace polynomials and hence generalize previous works in the literature that employ trace polynomials. The Reed-Solomon codes are over 𝔽_q^ℓ and have redundancy r = n-k ≥ q^m, 1≤ m≤ℓ, where n and k are the code length and dimension, respectively. In particular, for one erasure, we show that our schemes can achieve optimal repair bandwidths whenever n=q^ℓ and r = q^m, for all 1 ≤ m ≤ℓ. For two erasures, our schemes use the same bandwidth per erasure as the single erasure schemes, for ℓ/m is a power of q, and for ℓ=q^a, m=q^b-1>1 (a ≥ b ≥ 1), and for m≥ℓ/2 when ℓ is even and q is a power of two.
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