We establish the existence of optimal entanglement-assisted quantum [Formula: see text] codes for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. These codes are obtained from quaternary Hermitian linear complementary dual codes. We also give some observations on the largest minimum weights.
We establish the existence of optimal entanglement-assisted quantum [Formula: see text] codes for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. These codes are obtained from quaternary Hermitian linear complementary dual codes. We also give some observations on the largest minimum weights.
The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s + 5, 3, 16s + 3; 21s + 2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code.
The objective of this thesis is to find suboptimal and optimal parameters from classical codes and import them into entanglement-assisted quantum codes. The thesis begins by introducing classical error correction, followed by a detailed introduction to quantum computing. Topics that are discussed in the introduction include qubits, quantum phenomena, such as superposition and entanglement, and quantum gates/circuits. The thesis then reviews the basics of quantum error correction and provides Shor's code to reinforce the reader's understanding. Subsequently, the formalism of stabilizer codes is thoroughly examined. We then explain the generalized concept of stabilizer codes which is entanglement-assisted quantum codes. They do not require generators to satisfy the commutativity property. Rather, they utilize the usage of ebits to resolve the anti-commutativity constraint. Next, the thesis explains quaternary field and then the Java program implemented to find the optimal parameters. Lastly, the thesis concludes with presenting the parameters of the new codes that were obtained throughout the research. We have found the suboptimal largest distance for quaternary hermitian linear complementary dual codes that can be imported as entanglement-assisted quantum error correction for parameters [22, 9, 9 or 10]₄, [22, 12, 7 or 8]₄, [23, 8, 11 or 12]₄, [23, 10, 9 or 10]₄, [23, 13, 7 or 8]₄, [24, 10, 10 or 11]₄, [24, 11, 9 or 10]₄, [24, 14, 7 or 8]₄, [25, 12, 9 or 10]₄, [25, 13, 8 or 9]₄, as well as the optimal largest distance for [17, 11, 5]₄ and [17, 13, 3]₄.
We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.
Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.
We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corollaries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.
Abstract We introduce quaternary modified four μ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four μ μ -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.