Repetition Coding Research Articles

Investigation on repetition-coding and space-time-block-coding for indoor optical wireless communications employing beam shaping based on orbital angular momentum modes.

In this paper, we propose a novel beam shaping technique based on orbital angular momentum (OAM) modes for indoor optical wireless communications (OWC). Furthermore, we investigate two spatial diversity techniques, namely repetition-coding (RC) and Alamouti-type orthogonal space-time-block-coding (STBC) for indoor OWC employing the new beam shaping technique. The performance of both diversity schemes is systematically analyzed and compared under different beam shaping techniques using different OAM modes with different power ratios of the modes. It is shown that both RC and STBC can improve the system performance and effective coverage and RC outperforms STBC in all the beam shaping techniques regardless of the power ratios of the different modes. In addition, to further understand the performance of RC and STBC schemes against the signal delays induced during OAM mode conversion, the system tolerance of the two schemes to the delay interval is investigated with different OAM mode-based beam shaping techniques. Numerical results show that higher resistance to the delay interval can be achieved in STBC scheme. The advantage is more obvious when employing OAM0 and OAM1 based beam shaping technique.

Fault-tolerant fidelity based on few-qubit codes: parity-check circuits for biased error operators

In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e.~short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error operators are identified. Using the Steane code, when error operators are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achieved with the physical error rate of $10^{-5}$, which is much higher than the physical error rate of $10^{-9}$ for depolarising errors.

Coded Gradient Aggregation: A Tradeoff Between Communication Costs at Edge Nodes and at Helper Nodes

Increasing amount of data generated at edge nodes and quest for privacy have resulted in learning at the edge. Computations are performed at edge devices and outputs are communicated to a central node for updating the model. The edge nodes are available intermittently and are connected via low-bandwidth links. The edge nodes communicate local gradients to helper nodes, and these helpers forward messages to the central node after possible aggregation. Recently, schemes using repetition codes and maximum-distance-separable (MDS) codes, respectively known as aligned repetition coding (ARC) and aligned MDS coding (AMC) schemes, were proposed. It was observed that the communication cost at edge nodes becomes optimal in the AMC scheme, at the expense of an increased cost of communication incurred by helpers. An upper bound on the communication cost at helpers for the AMC scheme was known in literature. In this paper, a tradeoff between communication costs at edge nodes and at helper nodes is established with the help of newly proposed pyramid scheme. The scheme makes use of well-known class of pyramid codes, thus expanding the realm of application of locally repairable codes to distributed learning. The communication costs both at helper nodes and at edge nodes are exactly characterized. Using the developed technique, the exact communication cost at helper nodes can be computed for the AMC scheme as well. Next, we come up with a technique to improve the aggregation strategy of both pyramid and AMC schemes, that yields significant reduction in communication cost at helpers without changing parameters of the code used by edges. Finally, we present a greedy algorithm to improve the aggregation strategy of the ARC scheme, achieving significantly reduced communication cost at helpers.

On Some Capacity-Achieving Fractional Repetition Codes

Modern vehicle-based distributed storage networks need to provide efficient data recovery services due to the high mobility nature of storage vehicles. Compared with the simple replication scheme, erasure codes have been increasingly deployed in practical systems to achieve better storage performance. Specifically, fractional repetition (FR) codes are a class of repair efficient erasure codes characterized by the features of uncoded exact repair and minimum repair bandwidth. In this paper, we consider the capacity of FR codes, which is the maximum amount of information that can be obtained by any user contacting <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> out of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> storage nodes in the code based system. We present several families of FR codes that achieve a tight upper bound on the FR capacity, which are derived from combinatorial designs. The proposed constructions extend the parameter values of capacity-achieving FR codes to larger sets, and the resulting codes are of practical interest for vehicle-based distributed storage systems.

Limitations on Transversal Gates for Hypergraph Product Codes

In a fault tolerant quantum computer, quantum codes are expected to serve the conflicting purposes of protecting quantum information while also allowing that information to be manipulated by fault-tolerant gates. We introduce a new technique for placing limitations on such gates, and apply this technique to a class of quantum codes known as hypergraph product codes contained within the vertical sector. These codes are constructed from input which is a pair of classical linear codes, and generalize the Kitaev surface code which is the hypergraph product of classical repetition codes. We provide a necessary condition on these input codes, under which the resulting hypergraph product code has transversal gates limited to the Clifford group. We conjecture that this condition is satisfied by all <inline-formula> <tex-math notation="LaTeX">$[n,k,d]$ </tex-math></inline-formula> Gallagher codes with <inline-formula> <tex-math notation="LaTeX">$d\ge 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k\le n/2$ </tex-math></inline-formula>. This work is a generalization of an argument due to Bravyi and K&#x00F6;nig, and we also conjecture this is a refinement of the recent notion of disjointness due to Jochym-O&#x2019;Connor <i>et al.</i>

Performance Comparison of Repetition Coding MIMO Optical Wireless Communications with Distinct Light Beams.

In current optical wireless communications (OWC) oriented research works, the various multiple input multiple output (MIMO) techniques are introduced and utilized to enhance the coverage performance. Objectively, this Lambertian light beam based MIMO research paradigm neglects the light beam diversity and the potential performance gains. In this work, the distinct non-Lambertian light beams of commercially available light emitting diodes (LEDs) were adopted to configure MIMO OWC links. Specifically, the homogenous and heterogeneous non-Lambertian MIMO configurations were constituted in typical indoor scenarios. Moreover, applying the repetition coding (RC) MIMO algorithm with low complexity, a spatial coverage performance comparison was made between the several above mentioned non-Lambertian configurations and the well-discussed Lambertian MIMO configuration. Numerical results illustrate that the homogeneous NSPW light beam configuration could provide a more than 30 dB average signal to noise ratio (SNR), while the achievable average SNR of the heterogeneous light beam configuration was up to 28.77 dB. On the other side, the counterpart of the Lambertian configuration achieved just about 27.00 dB. Objectively, this work paves the fundamental and essential way for the further design and optimization of MIMO OWC in this novel light beam dimension.

An Improved Bound and Singleton-Optimal Constructions of Fractional Repetition Codes

Fractional repetition (FR) codes are a class of repair efficient erasure codes that can recover a failed storage node with both optimal repair bandwidth and complexity. In this paper, we study the minimum distance of FR codes, which is the smallest number of nodes whose failure leads to the unrecoverable loss of the stored file. We derive a new upper bound on the minimum distance of FR codes, which is tighter than the Singleton bound and a Singleton-like bound that takes locality into account. Based on regular graphs and combinatorial designs, several families of FR codes with optimal minimum distance are obtained.

Space-Time Block Coded Spatial Modulation for Indoor Visible Light Communications

Visible light communication (VLC) has been recognized as a promising technology for handling the continuously increasing quality of service and connectivity requirements in modern wireless communications, particularly in indoor scenarios. In this context, the present work considers the integration of two distinct modulation schemes, namely spatial modulation (SM) with space time block codes (STBCs), aiming at improving the overall VLC system reliability. Based on this and in order to further enhance the achievable transmission data rate, we integrate quasi-orthogonal STBC (QOSTBC) with SM, since relaxing the orthogonality condition of OSTBC ultimately provides a higher coding rate. Then, we generalize the developed results to any number of active light-emitting diodes (LEDs) and any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -ary pulse amplitude modulation size. Furthermore, we derive a tight and tractable upper bound for the corresponding bit error rate (BER) by considering a simple two-step decoding procedure to detect the indices of the transmitting LEDs and then decode the signal domain symbols. Notably, the obtained results demonstrate that QOSTBC with SM enhances the achievable BER compared to SM with repetition coding (RC-SM). Finally, we compare STBC-SM with both multiple active SM (MASM) and RC-SM in terms of the achievable BER and overall data rate, which further justifies the usefulness of the proposed scheme.

Parity-Check Coding Transmit Diversity for Wireless Communications With High Mobility

Highmobility is an important scenario for the 5G wireless communications. However, the high mobility results in serious time-varying fading, bringing challenges to the reliabilities of wireless transmission. Diversity technique is an essential method to overcome the effect of fading. It is noted that an LDPC (low-density parity-check) code is a concatenation of single parity-check codes (SPC) and repetition codes (REP) by a random interleaver, and a REP can be viewed as a typical example for achieving time/frequency diversity. Thus, it is appealing to jointly consider LDPC codes and diversity techniques. This paper presents parity-check coding transmit diversity (PCTD) schemes based on half-rate invertible LDPC codes, which are capable of achieving both diversity and coding gains from the time-varying fading channels. Besides, we investigate two types of PCTD schemes, and take into consideration two transmission modes, which are single-carrier (SC) and orthogonal frequency division multiplexing (OFDM) transmission modes. A chase combining based message passing algorithm (CC-MPA) is proposed for the proposed schemes, which can take full advantages of signal processing and coding gains. Furthermore, the Shannon limits of time-varying fading channels with BPSK signaling are deduced. Simulation results show that the proposed PCTD with CC-MPA algorithm can provide better BER performances than the classical time-domain diversity scheme in time-varying fading channels, which is appealing for the high mobility communications.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

Bounds on the covering radius of repetition code in Z2 Z6

In this paper, the covering radius of codes over R = Z 2 Z 6 with different weight are discussed. The block repetition codes over R is defined and the covering radius for block repetition codes R are obtained.

Free Ride on LDPC Coded Transmission

In this paper, we formulate the problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">accessible capacity</i> , is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryless channel, the accessible capacity can be characterized as the difference between the channel capacity and the mutual information rate of the coded payload link, which can be numerically evaluated for very short payload codes. For a general payload code, we present a simple lower bound on the accessible capacity, given by the channel capacity minus the coding rate of the payload code. We then focus on the scenarios where low-density parity-check (LDPC) codes are implemented for the payload link. We propose to transmit extra bits by random superposition for encoding, and exhaustive search (with the aid of statistical learning) for decoding. We further propose, by establishing an auxiliary channel (called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">syndrome channel</i> ) induced from “zero-forcing” over the binary field, to transmit extra bits with structured codes such as repetition codes and first-order Reed-Muller (RM) codes. Numerical results show that up to 60 extra bits can be reliably transmitted along with a rate-1/2 LDPC code of length 8064.

Bounds on the covering radius of repetition code in Z2 Z6

In this paper, the covering radius of codes over R = Z 2 Z 6 with different weight are discussed. The block repetition codes over R is defined and the covering radius for block repetition codes R are obtained.

Private and Secure Distributed Matrix Multiplication Schemes for Replicated or MDS-Coded Servers

In this paper, we study the problem of \emph{private and secure distributed matrix multiplication (PSDMM)}, where a user having a private matrix $A$ and $N$ non-colluding servers sharing a library of $L$ ($L>1$) matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$, for which the user wishes to compute $AB^{(\theta)}$ for some $\theta\in [0, L)$ without revealing any information of the matrix $A$ to the servers, and keeping the index $\theta$ private to the servers. Previous work is limited to the case that the shared library (\textit{i.e.,} the matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$) is stored across the servers in a replicated form and schemes are very scarce in the literature, there is still much room for improvement. In this paper, we propose two PSDMM schemes, where one is limited to the case that the shared library is stored across the servers in a replicated form but has a better performance than state-of-the-art schemes in that it can achieve a smaller recovery threshold and download cost. The other one focuses on the case that the shared library is stored across the servers in an MDS-coded form, which requires less storage in the servers. The second PSDMM code does not subsume the first one even if the underlying MDS code is degraded to a repetition code as they are totally two different schemes.

Demonstration of a 2 × 2 MIMO-UWOC system with large spot against air bubbles

In order to reduce turbulence-induced scintillation and deal with alignment problems, a 2×2 multiple-input multiple-output (MIMO) underwater wireless optical communication (UWOC) system is proposed and experimentally demonstrated. With help of the large divergence angle of light beams and large field of view (FOV) of the detectors, the effect of high-density air bubbles is greatly eliminated. Simulation and experimental results confirm that, in most intensity-modulation/direct-detection (IM/DD) MIMO-UWOC systems, the repetition coding (RC) scheme performs better than the space-time block coding (STBC) scheme. In a 50 m swimming pool, the maximum horizontal offset can reach 97.9 cm, which is 421% and 192% higher than that of STBC multiple-input single-output (MISO) and RC-MISO/STBC-MIMO schemes, respectively. With a data rate of 233 Mbps and a transmission distance of 50 m, the large detection range can meet a variety of underwater wireless communication requirements. The experiment indicates that, when the difference in the transmission distance between the two optical signals is higher than 1 m, the bit error rate (BER) of the RC scheme increases sharply, while the BER of the STBC scheme is stable. The MIMO coding scheme needs to be selected according to the actual application environment.

Optimizing Stabilizer Parities for Improved Logical Qubit Memories.

We study variants of Shor's code that are adept at handling single-axis correlated idling errors, which are commonly observed in many quantum systems. By using the repetition code structure of the Shor's code basis states, we calculate the logical channel applied to the encoded information when subjected to coherent and correlated single qubit idling errors, followed by stabilizer measurement. Changing the signs of the stabilizer generators allows us to change how the coherent errors interfere, leading to a quantum error-correcting code which performs as well as a classical repetition code of equivalent distance against these errors. We demonstrate a factor of 3.78±1.20 improvement of the logical T2^{*} in a distance-3 logical qubit implemented on a trapped-ion quantum computer. Even-distance versions of our Shor-code variants are decoherence-free subspaces and fully robust to identical and independent coherent idling noise.

Improving readout in quantum simulations with repetition codes

Near term quantum computers suffer from the presence of different noise sources. In order to mitigate for this effect and acquire results with significantly better accuracy, there is the urge of designing efficient error correction or error mitigation schemes. The cost of such techniques is usually high in terms of resource requirements, either in hardware or at the algorithmic level. In this work, we follow a pragmatic approach and we use repetition codes as scalable schemes with the potential to provide more accurate solutions to problems of interest in quantum chemistry and physics. We investigate different repetition code layouts and we propose a circular repetition scheme with connectivity requirements that are native on IBM Quantum hardware. We showcase our approach in multiple IBM Quantum devices and validate our results using a simplified theoretical noise model. We highlight the effect of using the proposed scheme in an electronic structure variational quantum eigensolver calculation and in the simulation of time evolution for a quantum Ising model.

Quantum repetition codes as building blocks of large-period discrete time crystals

Discrete time crystals (DTCs) are nonequilibrium phases of matter with exotic observable dynamics. Among their remarkable features is their response to a periodic drive at a fraction of its frequency. Current successful experiments are however only limited to realizing DTCs with period-doubling and period-tripling observable dynamics, forming only a very small subset of DTC phases. Creating larger periodic DTCs in the lab remains a longstanding challenge, yet it is necessary for developing the technological applications of DTCs, e.g., as a quantum memory for highly-entangled qubits, or exploring interesting features beyond subharmonic dynamics, e.g., condensed matter phenomena in the time domain. By highlighting the connection between DTCs and quantum error correction, we devise a general and realistic scheme for building DTCs exhibiting any large period observable dynamics, which is observable even at sufficiently small system sizes. Our proposal uses an array of spin-1/2 chains to simulate a repetition code at the hardware level, which has essential properties to realize robust observable dynamics. It is readily implemented with existing superconducting or trapped-ion quantum processors, making new families of DTCs experimentally accessible in the immediate future.

Quantum Secure Direct Communication with Mutual Authentication using a Single Basis

In this paper, we propose a new theoretical scheme for quantum secure direct communication (QSDC) with user authentication. Different from the previous QSDC protocols, the present protocol uses only one orthogonal basis of single-qubit states to encode the secret message. Moreover, this is a one-time and one-way communication protocol, which uses qubits prepared in a randomly chosen arbitrary basis, to transmit the secret message. We discuss the security of the proposed protocol against some common attacks and show that no eaves-dropper can get any information from the quantum and classical channels. We have also studied the performance of this protocol under realistic device noise. We have executed the protocol in IBMQ Armonk device and proposed a repetition code based protection scheme that requires minimal overhead.

Robust Rateless Spatially Coupled Repeat-Accumulate-Repeat Multi-User Codes on IDMA Systems

In this paper, we design a kind of robust rateless multi-user code, which is called spatially coupled repeat-accumulate-repeat (SC-RA-R) code, for the interleave-division multiple-access (IDMA) system. The code is constructed by serially concatenating a fixed outer spatially coupled repeat-accumulate (SC-RA) code with an adjustable inner irregular repetition code. It achieves rateless property by simply adjusting the parameters of the inner repetition code. The IDMA system with the proposed SC-RA-R multi-user codes can adapt to varying channel qualities and varying number of users via rateless property while keeping encoder and decoder implementation unchanged. Analytical extrinsic information transfer (EXIT) functions are derived to determine belief propagation (BP) decoding thresholds of SC-RA-R codes on the IDMA system. The optimization expression is also given under the constraint that repeat parameters are monotonically non-decreasing with the sum rate decreasing. The numerical results show that the maximum achievable sum rates of the SC-RA-R codes can be close to the Shannon bound for arbitrary channel qualities. Besides, our proposed SC-RA-R codes have better BP thresholds and bit-error-rate (BER) performances than the conventional optimal parallel-concatenate codes (PCC) and the optimal partially repeated spatially coupled low-density parity-check (PR-SC-LDPC) codes on IDMA systems.