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Integrated Performance of Next Generation High Data Rate Receiver and AR4JA LDPC Codec for Space Communications
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Optimizing quasi-cyclic spatially coupled LDPC codes by eliminating harmful objects
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Efficient Routing of Quantum LDPC Codes on Programmable 2D Toric Architectures
Quantum low-density parity-check codes are promising candidates towards scalable fault-tolerant quantum computation. Among these, bivariate bicycle (BB) codes offer superior encoding rates and large code distance compared to surface codes. However, their requirement on long-range stabilizer measurements poses significant challenges for implementation on realistic hardware with limited connectivity, such as superconducting circuit platforms. In this work, we introduce a novel hardware-software co-design that leverages a programmable communication network architecture to address these limitations. Our approach utilizes a 2D toric network of oscillators as a flexible communication fabric linking qubits at each site. Such architecture significantly reduces the number of long-range couplers required from O ( n ) to O (√ n ). Dual-rail qubits, along with native gates including Swap-Wait-Swap gates and beamsplitter SWAPs, ensure that long-range two-qubit gates can be executed with high fidelity and low latency. To further enhance performance, our qubit layout and routing algorithm utilize symmetries of the codes and enable maximum parallelism for long-range two-qubit gates, maintaining a low syndrome extraction cycle duration and scalability over the code length. We perform circuit-level simulation with realistic noise modeling based on experimental hardware parameters, observing an logical error rate per logical qubit per cycle of 3.06% for [[18,4,4]] BB code, 2.6× less than the existing experimental result. These findings provide a practical roadmap and identify key technological advancements needed to achieve low-overhead fault-tolerant quantum computing at scale.
Soft-Decision-Data Reshuffle to Mitigate Pulsed Radio Frequency Interference Impact on Low-Density-Parity-Check Code Performance
This presentation briefly discusses a research effort on mitigation techniques of pulsed radio frequency interference (RFI) on a Low-Density-Parity-Check (LDPC) code. This problem is of considerable interest in the context of providing reliable communications to the space vehicle which might suffer severe degradation due to pulsed RFI sources such as large radars. The LDPC code is one of modern forward-error-correction (FEC) codes which have the decoding performance to approach the Shannon Limit. The LDPC code studied here is the AR4JA (2048, 1024) code recommended by the Consultative Committee for Space Data Systems (CCSDS) and it has been chosen for some spacecraft design. Even though this code is designed as a powerful FEC code in the additive white Gaussian noise channel, simulation data and test results show that the performance of this LDPC decoder is severely degraded when exposed to the pulsed RFI specified in the spacecraft s transponder specifications. An analysis work (through modeling and simulation) has been conducted to evaluate the impact of the pulsed RFI and a few implemental techniques have been investigated to mitigate the pulsed RFI impact by reshuffling the soft-decision-data available at the input of the LDPC decoder. The simulation results show that the LDPC decoding performance of codeword error rate (CWER) under pulsed RFI can be improved up to four orders of magnitude through a simple soft-decision-data reshuffle scheme. This study reveals that an error floor of LDPC decoding performance appears around CWER=1E-4 when the proposed technique is applied to mitigate the pulsed RFI impact. The mechanism causing this error floor remains unknown, further investigation is necessary.
Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations
We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians 𝐻 0 defined by ⟦𝑁,𝐾,𝑑⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance 𝑑 ≥ 𝑐log (𝑁), implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by 𝐻 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the 𝑂(1) smallest eigenvalues of 𝐻 0 . The band originating from the smallest eigenvalue has 2 𝐾 states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth 𝛿 =𝐶𝑁𝑒 −Θ(𝑑) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.
Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery and More
Low density parity check (LDPC) codes with iterative decoding based on belief propagation achieve astonishing error performance close to Shannon limit. No algebraic or geometric method for constructing these codes has been reported and they are largely generated by computer search. As a result, encoding of long LDPC codes is in general very complex. This paper presents two classes of high rate LDPC codes whose constructions are based on finite Euclidean and projective geometries, respectively. These classes of codes a.re cyclic and have good constraint parameters and minimum distances. Cyclic structure adows the use of linear feedback shift registers for encoding. These finite geometry LDPC codes achieve very good error performance with either soft-decision iterative decoding based on belief propagation or Gallager's hard-decision bit flipping algorithm. These codes can be punctured or extended to obtain other good LDPC codes. A generalization of these codes is also presented.
Optimal Codes for the Burst Erasure Channel
Deep space communications over noisy channels lead to certain packets that are not decodable. These packets leave gaps, or bursts of erasures, in the data stream. Burst erasure correcting codes overcome this problem. These are forward erasure correcting codes that allow one to recover the missing gaps of data. Much of the recent work on this topic concentrated on Low-Density Parity-Check (LDPC) codes. These are more complicated to encode and decode than Single Parity Check (SPC) codes or Reed-Solomon (RS) codes, and so far have not been able to achieve the theoretical limit for burst erasure protection. A block interleaved maximum distance separable (MDS) code (e.g., an SPC or RS code) offers near-optimal burst erasure protection, in the sense that no other scheme of equal total transmission length and code rate could improve the guaranteed correctible burst erasure length by more than one symbol. The optimality does not depend on the length of the code, i.e., a short MDS code block interleaved to a given length would perform as well as a longer MDS code interleaved to the same overall length. As a result, this approach offers lower decoding complexity with better burst erasure protection compared to other recent designs for the burst erasure channel (e.g., LDPC codes). A limitation of the design is its lack of robustness to channels that have impairments other than burst erasures (e.g., additive white Gaussian noise), making its application best suited for correcting data erasures in layers above the physical layer. The efficiency of a burst erasure code is the length of its burst erasure correction capability divided by the theoretical upper limit on this length. The inefficiency is one minus the efficiency. The illustration compares the inefficiency of interleaved RS codes to Quasi-Cyclic (QC) LDPC codes, Euclidean Geometry (EG) LDPC codes, extended Irregular Repeat Accumulate (eIRA) codes, array codes, and random LDPC codes previously proposed for burst erasure protection. As can be seen, the simple interleaved RS codes have substantially lower inefficiency over a wide range of transmission lengths.
Accumulate repeat accumulate codes
In this paper we propose an innovative channel coding scheme called 'Accumulate Repeat Accumulate codes' (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of Low Density Parity Check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded Repeat Accumulate (RA) code or as precoded Irregular Repeat Accumulate (IRA) code, where simply an accumulator is chosen as a precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known LDPC codes with the dame maximum node degree. Furthermore by puncturing the accumulators any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation that allows for high speed decoder implementation.
Bilayer Protograph Codes for Half-Duplex Relay Channels
Direct to Earth return links are limited by the size and power of lander devices. A standard alternative is provided by a two-hops return link: a proximity link (from lander to orbiter relay) and a deep-space link (from orbiter relay to Earth). Although direct to Earth return links are limited by the size and power of lander devices, using an additional link and a proposed coding for relay channels, one can obtain a more reliable signal. Although significant progress has been made in the relay coding problem, existing codes must be painstakingly optimized to match to a single set of channel conditions, many of them do not offer easy encoding, and most of them do not have structured design. A high-performing LDPC (low-density parity-check) code for the relay channel addresses simultaneously two important issues: a code structure that allows low encoding complexity, and a flexible rate-compatible code that allows matching to various channel conditions. Most of the previous high-performance LDPC codes for the relay channel are tightly optimized for a given channel quality, and are not easily adapted without extensive re-optimization for various channel conditions. This code for the relay channel combines structured design and easy encoding with rate compatibility to allow adaptation to the three links involved in the relay channel, and furthermore offers very good performance. The proposed code is constructed by synthesizing a bilayer structure with a pro to graph. In addition to the contribution to relay encoding, an improved family of protograph codes was produced for the point-to-point AWGN (additive white Gaussian noise) channel whose high-rate members enjoy thresholds that are within 0.07 dB of capacity. These LDPC relay codes address three important issues in an integrative manner: low encoding complexity, modular structure allowing for easy design, and rate compatibility so that the code can be easily matched to a variety of channel conditions without extensive re-optimization. The main problem of half-duplex relay coding can be reduced to the simultaneous design of two codes at two rates and two SNRs (signal-to-noise ratios), such that one is a subset of the other. This problem can be addressed by forceful optimization, but a clever method of addressing this problem is via the bilayer lengthened (BL) LDPC structure. This method uses a bilayer Tanner graph to make the two codes while using a concept of "parity forwarding" with subsequent successive decoding that removes the need to directly address the issue of uneven SNRs among the symbols of a given codeword. This method is attractive in that it addresses some of the main issues in the design of relay codes, but it does not by itself give rise to highly structured codes with simple encoding, nor does it give rate-compatible codes. The main contribution of this work is to construct a class of codes that simultaneously possess a bilayer parity- forwarding mechanism, while also benefiting from the properties of protograph codes having an easy encoding, a modular design, and being a rate-compatible code.
Accumulate-Repeat-Accumulate-Accumulate-Codes
Inspired by recently proposed Accumulate-Repeat-Accumulate (ARA) codes [15], in this paper we propose a channel coding scheme called Accumulate-Repeat-Accumulate-Accumulate (ARAA) codes. These codes can be seen as serial turbo-like codes or as a subclass of Low Density Parity Check (LDPC) codes, and they have a projected graph or protograph representation; this allows for a high-speed iterative decoder implementation using belief propagation. An ARAA code can be viewed as a precoded Repeat-and-Accumulate (RA) code with puncturing in concatenation with another accumulator, where simply an accumulator is chosen as the precoder; thus ARAA codes have a very fast encoder structure. Using density evolution on their associated protographs, we find examples of rate-lJ2 ARAA codes with maximum variable node degree 4 for which a minimum bit-SNR as low as 0.21 dB from the channel capacity limit can be achieved as the block size goes to infinity. Such a low threshold cannot be achieved by RA or Irregular RA (IRA) or unstructured irregular LDPC codes with the same constraint on the maximum variable node degree. Furthermore by puncturing the accumulators we can construct families of higher rate ARAA codes with thresholds that stay close to their respective channel capacity thresholds uniformly. Iterative decoding simulation results show comparable performance with the best-known LDPC codes but with very low error floor even at moderate block sizes.
Structured Low-Density Parity-Check Codes with Bandwidth Efficient Modulation
In this work, we study the performance of structured Low-Density Parity-Check (LDPC) Codes together with bandwidth efficient modulations. We consider protograph-based LDPC codes that facilitate high-speed hardware implementations and have minimum distances that grow linearly with block sizes. We cover various higher- order modulations such as 8-PSK, 16-APSK, and 16-QAM. During demodulation, a demapper transforms the received in-phase and quadrature samples into reliability information that feeds the binary LDPC decoder. We will compare various low-complexity demappers and provide simulation results for assorted coded-modulation combinations on the additive white Gaussian noise and independent Rayleigh fading channels.
Flexible Data and Frame Synchronization Structure for the LunaNet PNT Signal
A LunaNet Lunar Augmented Navigation Service (LANS) is being developed to enable a position, navigation, and timing service for future Lunar operations [1]. The signal includes two components. An in-phase data channel signal is spread by a 1.023 MCPS ranging code that provides a high-rate data message at 250 bps and is encoded by a strong Low Density Parity Check (LDPC) code. A pilot channel with a 5.115 MCPS spreading code is also provided. The pilot code is configured with a secondary (overlay) code that does not currently provide absolute time or frame Synchronization as is the case for L1C [2] [3]. This work shows the advantage of implementing an overlay structure that provides absolute time for the LunaNet LANS signal structure known as the Augmented Forward Signal (AFS). The LunaNet AFS structure was developed to service two classes of user receivers. The first class is a low-complexity user receiver that only receives the 1.023 MCPS signal and does not use the 5.115 MCPS pilot channel. For this class of user, a data frame Sync word is needed. The second class of receiver is a high-end receiver that can processes both data and pilot channels to take advantage of the higher chip rate pilot channel for enhanced robustness and improved accuracy. In the current draft LunaNet LANS AFS design, these users must employ the frame Sync word in the data channel and obtain absolute time after decoding the AFS navigation message [2]. The signal structure would greatly benefit from the addition of a pilot overlay structure that provides absolute time and robust frame Sync for high-end users as done for L1C [3]. To provide a more robust and interoperable AFS structure, this work summarizes a study and recommends alternatives for a new overlay code on the pilot channel that provides absolute time and a Sync word approach on the data channel. The overlay code and Sync word are designed to allow for flexible and robust data synchronization for both low- complexity and high-end user receivers. The new overlay code structure permits frame synchronization performance that is as good as or better than the L1 C signal, while enabling a determination of absolute time upon frame Sync to aid high-end assisted LANS AFS user receivers at low signal to noise levels. We also present the design of rate-matched 5G new radio (5GNR) LDPC codes that fit within the current 6000-symbol frame size along with a time of interval (TOI) word, frame ID (FID) word, and the remaining LunaNet AFS data message blocks. The paper describes and demonstrates robust frame Synchronization performance of the overlay code and Synch word approaches. The results are described in terms of probability of missed detection and probability of false alarm for a correct frame Synchronization at low Eb/No levels expected for decoding the TOI word and LDPC encoded data. Advantages of the proposed data Synchronization structure will be described along with use cases for low-end and high-end receivers. Practical implementation considerations will also be described.
Flexible Data and Frame Synchronization Structure for the LunaNet PNT Signal
A LunaNet Lunar Augmented Navigation Service (LANS) is being developed to enable a position, navigation, and timing service for future Lunar operations. The signal includes two components. An in-phase data channel signal is spread by a 1.023 MCPS ranging code that provides a high-rate data message at 250 bps and is encoded by a strong Low Density Parity Check (LDPC) code. A pilot channel with a 5.115 MCPS spreading code is also provided. The pilot code is configured with a secondary (overlay) code that does not currently provide absolute time or frame Synchronization as is the case for L1C. This work shows the advantage of implementing an overlay structure that provides absolute time for the LunaNet LANS signal structure known as the Augmented Forward Signal (AFS). The LunaNet AFS structure was developed to service two classes of user receivers. The first class is a low-complexity user receiver that only receives the 1.023 MCPS signal and does not use the 5.115 MCPS pilot channel. For this class of user, a data frame Sync word is needed. The second class of receiver is a high-end receiver that can processes both data and pilot channels to take advantage of the higher chip rate pilot channel for enhanced robustness and improved accuracy. In the current draft LunaNet LANS AFS design, these users must employ the frame Sync word in the data channel and obtain absolute time after decoding the AFS navigation message. The signal structure would greatly benefit from the addition of a pilot overlay structure that provides absolute time and robust frame Sync for high-end users as done for L1C. To provide a more robust and interoperable AFS structure, this work summarizes a study and recommends alternatives for a new overlay code on the pilot channel that provides absolute time and a Sync word approach on the data channel. The overlay code and Sync word are designed to allow for flexible and robust data synchronization for both low- complexity and high-end user receivers. The new overlay code structure permits frame synchronization performance that is as good as or better than the L1 C signal, while enabling a determination of absolute time upon frame Sync to aid high-end assisted LANS AFS user receivers at low signal to noise levels. We also present the design of rate-matched 5G new radio (5GNR) LDPC codes that fit within the current 6000-symbol frame size along with a time of interval (TOI) word, frame ID (FID) word, and the remaining LunaNet AFS data message blocks. The paper describes and demonstrates robust frame Synchronization performance of the overlay code and Synch word approaches. The results are described in terms of probability of missed detection and probability of false alarm for a correct frame Synchronization at low Eb/No levels expected for decoding the TOI word and LDPC encoded data. Advantages of the proposed data Synchronization structure will be described along with use cases for low-end and high-end receivers. Practical implementation considerations will also be described.
Maximum likelihood decoding analysis of Accumulate-Repeat-Accumulate Codes
Repeat-Accumulate (RA) codes are the simplest turbo-like codes that achieve good performance. However, they cannot compete with Turbo codes or low-density parity check codes (LDPC) as far as performance is concerned. The Accumulate Repeat Accumulate (ARA) codes, as a subclass of LDPC codes, are obtained by adding a pre-coder in front of RA codes with puncturing where an accumulator is chosen as a precoder. These codes not only are very simple, but also achieve excellent performance with iterative decoding. In this paper, the performance of these codes with (ML) decoding are analyzed and compared to random codes by very tight bounds. The weight distribution of some simple ARA codes is obtained, and through existing tightest bounds we have shown the ML SNR threshold of ARA codes approaches very closely to the performance of random codes. We have shown that the use of precoder improves the SNR threshold but interleaving gain remains unchanged with respect to RA code with puncturing.
Low-Density Parity-Check Codes as Stable Phases of Quantum Matter
Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error-correcting code, when does it define a stable gapped quantum phase of matter, whose ground-state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits “check soundness.” Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric-code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.
Diffusion Codes: Self-Correction from Small(er)-Set Expansion with Tunable Non-locality
Optimal constructions of classical LDPC codes can be obtained by choosing the Tanner graph uniformly at random among biregular graphs. We introduce a class of codes that we call ``diffusion codes'', defined by placing each edge connecting bits and checks on some graph, and acting on that graph with a random SWAP network. By tuning the depth of the SWAP network, we can tune a tradeoff between the amount of randomness -- and hence the optimality of code parameters -- and locality with respect to the underlying graph. For diffusion codes defined on the cycle graph, if the SWAP network has depth $\sim Tn$ with $T> n^{2β}$ for arbitrary $β>0$, then we prove that almost surely the Tanner graph is a lossless ``smaller set'' vertex expander for small sets up size $δ\sim \sqrt T \sim n^β$, with bounded bit and check degree. At the same time, the geometric size of the largest stabilizer is bounded by $\sqrt T$ in graph distance. We argue, based on physical intuition, that this result should hold more generally on arbitrary graphs. By taking hypergraph products of these classical codes we obtain quantum LDPC codes defined on the torus with smaller-set boundary and co-boundary expansion and the same expansion/locality tradeoffs as for the classical codes. These codes are self-correcting and admit single-shot decoding, while having the geometric size of the stabilizer growing as an arbitrarily small power law. Our proof technique establishes mixing of a random SWAP network on small subsystems at times scaling with only the subsystem size, which may be of independent interest.