Related papers: Quantum $(r,δ)$-locally recoverable codes

Quantum $(r,δ)$-locally recoverable codes

URL: http://arxiv.org/abs/2412.16590v1
Date: Sat, 21 Dec 2024 11:45:32 GMT
Title: Quantum $(r,δ)$-locally recoverable codes
Authors: Carlos Galindo, Fernando Hernando, Helena Martín-Cruz, Ryutaroh Matsumoto,
Abstract summary: A classical $(r,delta)$-locally recoverable code is an error-correcting code such that, for each coordinate $c_i$ of a codeword, there exists a set of at most $r+ delta -1$ coordinates containing $c_i$. These codes are useful for avoiding loss of information in large scale distributed and cloud storage systems.
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Abstract: A classical $(r,\delta)$-locally recoverable code is an error-correcting code such that, for each coordinate $c_i$ of a codeword, there exists a set of at most $r+ \delta -1$ coordinates containing $c_i$ which allow us to correct any $\delta -1$ erasures in that set. These codes are useful for avoiding loss of information in large scale distributed and cloud storage systems. In this paper, we introduce quantum $(r,\delta)$-locally recoverable codes and give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,\delta)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{\perp_s}$. When a quantum stabilizer code comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,\delta)$-local recoverability. A Singleton-like bound is stated in this case and examples of optimal stabilizer $(r,\delta)$-locally recoverable codes are given.

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