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1. On the Duality Between the BSC and Quantum PSC

   https://doi.org/10.1109/ISIT45174.2021.9518034  

   Rengaswamy, Narayanan; Pfister, Henry D. (July 2021, 2021 IEEE International Symposium on Information Theory (ISIT))

   null (Ed.)

   In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)] developed a general theory of channel duality for classical-input quantum-output channels. His result shows that a number of well-known duality results for linear codes on the binary erasure channel can be extended to general classical channels at the expense of using dual problems which are intrinsically quantum mechanical. One special case of this duality is a connection between coding for error correction on the quantum pure-state channel (PSC) and coding for wiretap secrecy on the classical binary symmetric channel (BSC). Similarly, coding for error correction on the BSC is related to wire-tap secrecy on the PSC. While this result has important implications for classical coding, the machinery behind the general duality result is rather challenging for researchers without a strong background in quantum information theory. In this work, we leverage prior results for linear codes on PSCs to give an alternate derivation of the aforementioned special case by computing closed-form expressions for the performance metrics. The noted prior results include the optimality of square-root measurement for linear codes on the PSC and the Fourier duality of linear codes. 

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2. Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation

   https://doi.org/10.1109/Allerton63246.2024.10735275  

   Gökduman, Mert; Yao, Hanwen; Pfister, Henry D (September 2024, IEEE)

   Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape. 

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3. Channel coding at low capacity

   Fereydounian, Mohammad; Hassani, Hamed; Jamal, Vahid; Mahdavifar, Hessam (August 2023, IEEE journal on selected areas in information theory)

   Low-capacity scenarios have become increasingly important in the technology of the In- ternet of Things (IoT) and the next generation of wireless networks. Such scenarios require efficient and reliable transmission over channels with an extremely small capacity. Within these constraints, the state-of-the-art coding techniques may not be directly applicable. More- over, the prior work on the finite-length analysis of optimal channel coding provides inaccurate predictions of the limits in the low-capacity regime. In this paper, we study channel coding at low capacity from two perspectives: fundamental limits at finite length and code construc- tions. We first specify what a low-capacity regime means. We then characterize finite-length fundamental limits of channel coding in the low-capacity regime for various types of channels, including binary erasure channels (BECs), binary symmetric channels (BSCs), and additive white Gaussian noise (AWGN) channels. From the code construction perspective, we charac- terize the optimal number of repetitions for transmission over binary memoryless symmetric (BMS) channels, in terms of the code blocklength and the underlying channel capacity, such that the capacity loss due to the repetition is negligible. Furthermore, it is shown that capacity- achieving polar codes naturally adopt the aforementioned optimal number of repetitions. 

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4. New Dual Relationships for Error-Correcting Wiretap Codes

   https://doi.org/10.1109/ITW48936.2021.9611443  

   Shoushtari, Morteza; Harrison, Willie K. (October 2021, 2021 Information Theory Workshop (ITW))

   In this paper, we consider the equivocation of finite blocklength coset codes when used over binary erasure wiretap channels. We make use of the equivocation matrix in comparing codes that are suitable for scenarios with noisy channels for both the intended receiver and an eavesdropper. Equivocation matrices have been studied in the past only for the binary erasure wiretap channel model with a noiseless channel for the intended recipient. In that case, an exact relationship between the elements of equivocation matrices for a code and its dual code was identified. The majority of work on coset codes for wiretap channels only addresses the noise-free main channel case, and extensions to noisy main channels require multi-edge type codes. In this paper, we supply a more insightful proof for the noiseless main channel case, and identify a new dual relationship that applies when two-edge type coset codes are used for the noisy main channel case. The end result is that the elements of the equivocation matrix for a dual code are known precisely from the equivocation matrix of the original code according to fixed reordering patterns. Such relationships allow one to study the equivocation of codes and their duals in tandem, which simplifies the search for best and/or good finite blocklength codes. This paper is the first work that succinctly links the equivocation/error correction capabilities of dual codes for two-edge type coset coding over erasure-prone main channels. 

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5. Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback

   https://doi.org/10.1109/ISIT57864.2024.10619630  

   Yang, Hengjie; Wesel, Richard D (July 2024, IEEE)

   This paper presents new achievability bounds on the maximal achievable rate of variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC) at a fixed message size M=2^k . We provide bounds for two cases: The first case considers VLSF codes with possibly infinite decoding times and zero error probability. The second case limits the maximum (finite) number of decoding times and specifies a maximum tolerable probability of error. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first k message bits followed by random linear fountain parity bits decoded with a rank decoder. For VLSF codes with infinite decoding times, our new bound outperforms the state-of-the-art result for BEC by Devassy et al. in 2016. We show that the backoff from capacity reduces to zero as the erasure probability decreases, thus giving a negative answer to the open question Devassy et al. posed on whether the 23.4% backoff to capacity at k=3 is fundamental to all BECs. For VLSF codes with finite decoding times, numerical evaluations show that the systematic transmission followed by random linear fountain coding performs better than random linear coding in terms of achievable rates. 

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