Applications of Error Correcting Codes

Seong‐Joon Park, Kwak, Hee-Youl, Sang‐Hyo Kim +3 more

In communication and storage systems, error correction codes (ECCs) are pivotal in ensuring data reliability. As deep learning's applicability has broadened across diverse domains, there is a growing research focus on neural network-based decoders that outperform traditional decoding algorithms. Among these neural decoders, Error Correction Code Transformer (ECCT) has achieved the state-of-the-art performance, outperforming other methods by large margins. To further enhance the performance of ECCT, we propose two novel methods. First, leveraging the systematic encoding technique of ECCs, we introduce a new masking matrix for ECCT, aiming to improve the performance and reduce the computational complexity. Second, we propose a novel transformer architecture of ECCT called a double-masked ECCT. This architecture employs two different mask matrices in a parallel manner to learn more diverse features of the relationship between codeword bits in the masked self-attention blocks. Extensive simulation results show that the proposed double-masked ECCT outperforms the conventional ECCT, achieving the state-of-the-art decoding performance with significant margins.

Ivan B. Djordjević

E. Desurvire

This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes , as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code , which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t , both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code , which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code , based on the principle of majority logic . The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.

Yuma Yamashita, Hiroyuki Endo, Shingo Fujita +3 more

Quantum key distribution (QKD) is a technology to securely share keys against any attack physically permitted, with the principle of quantum mechanics. In recent years, the satellite QKD, which employs artificial satellites as trusted mobile nodes, has been attracting attention in order to overcome the bottleneck of transmission distance. However, in the satellite QKD, quality degradation due to atmospheric effects is expected, as in ordinary satellite laser communications. Therefore, it is desirable to apply an error-correcting code (ECC) that has high error-correcting performance even under the atmospheric-induced effects to the error-correcting process of the satellite QKD. Therefore, in this paper, we examined the application of polar codes, which is known as an ECC with high error correction capability. First, in order to optimize the error correction efficiency, we propose a method to adaptively obtain an appropriate code rate for the received signal strength that changes momentarily due to atmospheric effects. Then, we compare the throughput performances with polar codes to it with low-density parity-check (LDPC) codes, with the numerical simulation assuming Bennett-Brassard 1984 protocol (BB84).

Richard W. Hamming