This PhD thesis addresses two current problems belonging to the fields of classical and quantum information theory. The first is the repair problem in distributed and cloud storage systems in the setting where multiple nodes may fail simultaneously. These codes are called (r,\delta)-locally recoverable codes. To handle it, we consider monomial-Cartesian codes, resulting in new (r,\delta)-locally recoverable codes. The other problem is the construction of good quantum error-correcting codes. Those we provide are quantum stabilizer codes derived from two families of classical codes: generalized (or twisted) monomial-Cartesian codes and evaluation codes at the roots of trace-depending polynomials. Both quantum code constructions have excellent parameters.
This PhD thesis addresses two current problems belonging to the fields of classical and quantum information theory. The first is the repair problem in distributed and cloud storage systems in the setting where multiple nodes may fail simultaneously. These codes are called (r,\delta)-locally recoverable codes. To handle it, we consider monomial-Cartesian codes, resulting in new (r,\delta)-locally recoverable codes. The other problem is the construction of good quantum error-correcting codes. Those we provide are quantum stabilizer codes derived from two families of classical codes: generalized (or twisted) monomial-Cartesian codes and evaluation codes at the roots of trace-depending polynomials. Both quantum code constructions have excellent parameters.
Classical ( r , δ ) -locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum ( r , δ ) -locally recoverable codes which are quantum error-correcting codes capable of correcting δ − 1 qudit erasures from sets of at most r + δ − 1 qudits. We give a necessary and sufficient condition for a quantum stabilizer code Q ( C ) to be ( r , δ ) -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 ⊥ s . When Q ( C ) 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 , δ ) -local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.
Optimal pure quantum (r,δ)-locally recoverable codes from matrix-product construction
Locally recoverable codes (LRCs) are classical error-correcting codes widely used in large-scale distributed and cloud storage systems. Quantum locally recoverable codes of locality (r,δ) (quantum (r,δ)-LRCs) are the quantum counterpart of classical (r,δ)-LRCs. They allow us to correct erasures at several positions using a trace-preserving quantum operation acting on qudits of a larger set of positions. Quantum (r,δ)-LRCs, Q(C), can be constructed from classical Euclidean (or Hermitian) dual-containing codes C, and their recovery abilities are upper bounded by the minimum distance of the Euclidean (or Hermitian) dual of those codes. Parameters and localities of pure quantum (r,δ)-LRCs satisfy a Singleton-like bound; codes attaining equality are referred to as optimal. We consider matrix-product codes (MPCs) C and give constituent (or defining) matrices and conditions on the constituent codes such that the codes C satisfy the conditions to provide quantum (r,δ)-LRCs. As a consequence, we are able to determine their locality and parameters. Furthermore, we determine families of optimal pure quantum (r,δ)-LRCs derived from them.
A classical linear code C of length n is said to have symbol locality (r,δ) if for any index j there exists a repair group Jj⊆{1,…,n} with j∈Jj and ∣Jj∣≤r+δ−1 such that any δ−1 or fewer erasures in Jj can be corrected by using codeword symbols only in Jj. Later it turned out that this way of defining r overestimates the number of necessary codeword symbols for multiple-erasure correction, and information locality was proposed to define r as the dimension of the punctured code of C onto Jj. Recently locality (r,δ) was proposed for quantum error-correcting codes by following the original definition of symbol locality (r,δ). We propose a quantum counterpart of the information locality for quantum stabilizer codes constructed by Hermitian orthogonality, and a linear algebraic procedure computing a smaller repair group predicted by the proposed information locality and simultaneously reducing the number of measured observables in decoding to its minimum possible value. Then we demonstrate that the previously proposed definition of quantum locality (r,δ) has the same drawback of overestimating the number of necessary codeword symbols for erasure correction by providing an explicit example of a quantum stabilizer code. Finally, we will give another example of a quantum stabilizer code constructed by Euclidean orthogonality and two different linear codes, with which a natural translation of the classical information locality into the quantum setting underestimates the number of necessary codeword symbols for erasure correction.
By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.
Quantum (r,δ)-Locally Recoverable BCH and Homothetic-BCH Codes
Quantum (r,δ)-locally recoverable codes ((r,δ)-LRCs) are the quantum version of classical (r,δ)-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum (r,δ)-LRC, Q(C), can be constructed from an (r,δ)-LRC, C, which is Euclidean or Hermitian dual-containing.
This article is devoted to studying how to get quantum (r,δ)-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum (r,δ)-LRCs which are optimal for the Singleton-like bound.
Quantum (r,δ)-locally recoverable codes are quantum error-correcting codes capable of correcting δ−1 qudit erasures within one subset of qudits of cardinality at most r+δ−1. In this paper, we introduce the more general framework of entanglement-assisted quantum (r,δ)-locally recoverable codes, assuming that the local recovery operation is assisted by receiver-held qudits that remain unaffected by erasures.
We establish necessary and sufficient conditions for these codes to satisfy this property. For codes derived from Hermitian or Euclidean constructions, we establish connections between entanglement-assisted quantum and classical notions of (r,δ)-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum (r,δ)-locally recoverable codes from several families of classical codes, including bivariate J-affine variety codes, BCH codes, and homothetic-BCH codes.