Calderbank and Shor demonstrated the existence of good quantum error-correcting codes. These codes encode k qubits into n qubits such that if any t qubits undergo arbitrary decoherence, the original state can be faithfully reconstructed. The asymptotic rate of these codes is given by $ k/n = 1 - 2H_{2}(2t/n) $, where $ H_{2}(p) $ is the binary entropy function. The paper also provides upper bounds on this rate. The authors show that quantum error-correcting codes can be constructed using classical error-correcting codes. They use the Hamming code as an example, demonstrating how it can be adapted to create a quantum code that corrects one error. The quantum code is constructed by mapping the Hamming code's codewords into a subspace of the quantum Hilbert space. This allows for the correction of both bit and phase errors. The paper also discusses the decoding process for quantum codes, showing that errors in any t qubits can be corrected by first correcting bit errors in the $ |c\rangle $ basis and then correcting bit errors in the $ |s\rangle $ basis. The decoding process is shown to be effective for a wide range of physical quantum channels, including those with independent decoherence processes. The authors also explore the relationship between classical and quantum error-correcting codes, showing that weakly self-dual codes can be used to achieve the Gilbert-Varshamov bound. This bound provides a lower limit on the rate of quantum error-correcting codes. Finally, the paper discusses the application of these codes to quantum channels, showing that they can achieve high fidelity transmission of quantum states. The paper also provides an upper bound on the classical information capacity of quantum channels, based on the Levitin-Holevo theorem. This bound is plotted in Figure 1, showing the asymptotic rate of the quantum codes versus the error rate of the channel.Calderbank and Shor demonstrated the existence of good quantum error-correcting codes. These codes encode k qubits into n qubits such that if any t qubits undergo arbitrary decoherence, the original state can be faithfully reconstructed. The asymptotic rate of these codes is given by $ k/n = 1 - 2H_{2}(2t/n) $, where $ H_{2}(p) $ is the binary entropy function. The paper also provides upper bounds on this rate. The authors show that quantum error-correcting codes can be constructed using classical error-correcting codes. They use the Hamming code as an example, demonstrating how it can be adapted to create a quantum code that corrects one error. The quantum code is constructed by mapping the Hamming code's codewords into a subspace of the quantum Hilbert space. This allows for the correction of both bit and phase errors. The paper also discusses the decoding process for quantum codes, showing that errors in any t qubits can be corrected by first correcting bit errors in the $ |c\rangle $ basis and then correcting bit errors in the $ |s\rangle $ basis. The decoding process is shown to be effective for a wide range of physical quantum channels, including those with independent decoherence processes. The authors also explore the relationship between classical and quantum error-correcting codes, showing that weakly self-dual codes can be used to achieve the Gilbert-Varshamov bound. This bound provides a lower limit on the rate of quantum error-correcting codes. Finally, the paper discusses the application of these codes to quantum channels, showing that they can achieve high fidelity transmission of quantum states. The paper also provides an upper bound on the classical information capacity of quantum channels, based on the Levitin-Holevo theorem. This bound is plotted in Figure 1, showing the asymptotic rate of the quantum codes versus the error rate of the channel.