Recovery Sets of Subspaces from a Simplex Code
CoRR(2024)
Abstract
Recovery sets for vectors and subspaces are important in the construction of
distributed storage system codes. These concepts are also interesting in their
own right. In this paper, we consider the following very basic recovery
question: what is the maximum number of possible pairwise disjoint recovery
sets for each recovered element? The recovered elements in this work are
d-dimensional subspaces of a k-dimensional vector space over GF(q). Each
server stores one representative for each distinct one-dimensional subspace of
the k-dimensional vector space, or equivalently a distinct point of PG(k-1,q).
As column vectors, the associated vectors of the stored one-dimensional
subspaces form the generator matrix of the [(q^k -1)/(q-1),k,q^k-1] simplex
code over GF(q). Lower bounds and upper bounds on the maximum number of such
recovery sets are provided. It is shown that generally, these bounds are either
tight or very close to being tight.
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Key words
Availability,distributed storage,recovery sets,subspaces
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