Repairing Reed-Solomon Codes via Subspace Polynomials

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 \mathbb {F}_{{q}^{\ell }} and have redundancy {{r}} = {{n}}-{{k}} \geq {{q}}^{{m}} , 1\leq {{m}}\leq \ell , 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}}^\ell and {{r}} = {{q}}^{{m}} , for all 1 \leq {{m}} \leq \ell . For two erasures, our schemes use the same bandwidth per erasure as the single erasure schemes, for \ell /{{m}} is a power of {{q}} , and for \ell ={{q}}^{{a}} , {{m}}={{q}}^{{b}}-1>1 ( {{a}} \geq {{b}} \geq 1 ), and for {{m}}\geq \ell /2 when \ell is even and {{q}} is a power of two.