TY - JOUR

T1 - Repairing Reed–Solomon Codes Evaluated on Subspaces

AU - Berman, Amit

AU - Buzaglo, Sarit

AU - Dor, Avner

AU - Shany, Yaron

AU - Tamo, Itzhak

N1 - Publisher Copyright:
IEEE

PY - 2022

Y1 - 2022

N2 - We consider the repair problem for Reed–Solomon (RS) codes, evaluated on an Fq-linear subspace U ⊆ Fqm of dimension d, where q is a prime power, m is a positive integer, and Fq is the Galois field of size q. For q > 2, we show the existence of a linear repair scheme for the RS code of length n = qd and codimension qs, s < d, evaluated on U, in which each of the n-1 surviving nodes transmits only r symbols of Fq, provided that ms ≥ d(m - r). For the case q = 2, we prove a similar result, with some restrictions on the evaluation linear subspace U. Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least 1/3) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau–Milenkovic to the range r < m - s, for a wide range of parameters.

AB - We consider the repair problem for Reed–Solomon (RS) codes, evaluated on an Fq-linear subspace U ⊆ Fqm of dimension d, where q is a prime power, m is a positive integer, and Fq is the Galois field of size q. For q > 2, we show the existence of a linear repair scheme for the RS code of length n = qd and codimension qs, s < d, evaluated on U, in which each of the n-1 surviving nodes transmits only r symbols of Fq, provided that ms ≥ d(m - r). For the case q = 2, we prove a similar result, with some restrictions on the evaluation linear subspace U. Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least 1/3) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau–Milenkovic to the range r < m - s, for a wide range of parameters.

KW - Bandwidth

KW - Codes

KW - Maintenance engineering

KW - Probabilistic logic

KW - Research and development

KW - Symbols

KW - Time complexity

UR - http://www.scopus.com/inward/record.url?scp=85130779743&partnerID=8YFLogxK

U2 - 10.1109/TIT.2022.3177903

DO - 10.1109/TIT.2022.3177903

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AN - SCOPUS:85130779743

SN - 0018-9448

SP - 1

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

ER -