Steane-Enlargement of Quantum Codes from the Hermitian Curve

In this paper, we study the construction of quantum codes by applying Steane-enlargement to codes from the Hermitian curve. We cover Steane-enlargement of both usual one-point Hermitian codes and of order bound improved Hermitian codes. In particular, the paper contains two constructions of quantum codes whose parameters are described by explicit formulae, and we show that these codes compare favourably to existing, comparable constructions in the literature.Comment: 11 page

On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets

In this work, we study quantum error-correcting codes obtained by using Steane-enlargement. We apply this technique to certain codes defined from Cartesian products previously considered by Galindo et al. in [4]. We give bounds on the dimension increase obtained via enlargement, and additionally give an algorithm to compute the true increase. A number of examples of codes are provided, and their parameters are compared to relevant codes in the literature, which shows that the parameters of the enlarged codes are advantageous. Furthermore, comparison with the Gilbert-Varshamov bound for stabilizer quantum codes shows that several of the enlarged codes match or exceed the parameters promised by the bound.Comment: 12 page

Quantum codes from two-point Hermitian codes

We explore the background on error-correcting codes, including linear codes and quantum codes from curves. Then we consider the parameters of quantum codes constructed from two-point Hermitian codes

Sparse Graph Codes for Quantum Error-Correction

We present sparse graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened version was resubmitted to IEEE Transactions on Information Theory Jan 20, 200

On nested code pairs from the Hermitian curve

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance.Comment: 28 page

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

Quantum codes from a new construction of self-orthogonal algebraic geometry codes

[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.G. McGuire was partially supported by Science Foundation Ireland Grant 13/IA/1914. The remainder authors were partially supported by the Spanish Government and the EU funding program FEDER, Grants MTM2015-65764-C3-2-P and PGC2018-096446-B-C22. F. Hernando and J. J. Moyano-Fernandez are also partially supported by Universitat Jaume I, Grant UJI-B2018-10.Hernando, F.; Mcguire, G.; Monserrat Delpalillo, FJ.; Moyano-Fernández, JJ. (2020). Quantum codes from a new construction of self-orthogonal algebraic geometry codes. 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