A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes

We present an algorithm for systematic encoding of Hermitian codes. For a Hermitian code defined over GF(q^2), the proposed algorithm achieves a run time complexity of O(q^2) and is suitable for VLSI implementation. The encoder architecture uses as main blocks q varying-rate Reed-Solomon encoders and achieves a space complexity of O(q^2) in terms of finite field multipliers and memory elements.Comment: 5 Pages, Accepted in IEEE International Symposium on Information Theory ISIT 200

Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Grobner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n^3) using Gaussian elimination to O(qn^2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE Transactions on Information Theory on Jan. 8, 201

Unitary space-time modulation via Cayley transform

A prevoiusly proposed method for communicating with multiple antennas over block fading channels is unitary space-time modulation (USTM). In this method, the signals transmitted from the antennas, viewed as a matrix with spatial and temporal dimensions, form a unitary matrix, i.e., one with orthonormal columns. Since channel knowledge is not required at the receiver, USTM schemes are suitable for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Previous results have shown that if suitably designed, USTM schemes can achieve full channel capacity at high SNR and, moreover, that all this can be done over a single coherence interval, provided the coherence interval and number of transmit antennas are sufficiently large, which is a phenomenon referred to as autocoding. While all this is well recognized, what is not clear is how to generate good performing constellations of (nonsquare) unitary matrices that lend themselves to efficient encoding/decoding. The schemes proposed so far either exhibit poor performance, especially at high rates, or have no efficient decoding algorithms. We propose to use the Cayley transform to design USTM constellations. This work can be viewed as a generalization, to the nonsquare case, of the Cayley codes that have been proposed for differential USTM. The codes are designed based on an information-theoretic criterion and lend themselves to polynomial-time (often cubic) near-maximum-likelihood decoding using a sphere decoding algorithm. Simulations suggest that the resulting codes allow for effective high-rate data transmission in multiantenna communication systems without knowing the channel. However, our preliminary results do not show a substantial advantage over training-based schemes

Simplified decoding techniques for linear block codes

Error correcting codes are combinatorial objects, designed to enable reliable transmission of digital data over noisy channels. They are ubiquitously used in communication, data storage etc. Error correction allows reconstruction of the original data from received word. The classical decoding algorithms are constrained to output just one codeword. However, in the late 50’s researchers proposed a relaxed error correction model for potentially large error rates known as list decoding. The research presented in this thesis focuses on reducing the computational effort and enhancing the efficiency of decoding algorithms for several codes from algorithmic as well as architectural standpoint. The codes in consideration are linear block codes closely related to Reed Solomon (RS) codes. A high speed low complexity algorithm and architecture are presented for encoding and decoding RS codes based on evaluation. The implementation results show that the hardware resources and the total execution time are significantly reduced as compared to the classical decoder. The evaluation based encoding and decoding schemes are modified and extended for shortened RS codes and software implementation shows substantial reduction in memory footprint at the expense of latency. Hermitian codes can be seen as concatenated RS codes and are much longer than RS codes over the same aphabet. A fast, novel and efficient VLSI architecture for Hermitian codes is proposed based on interpolation decoding. The proposed architecture is proven to have better than Kötter’s decoder for high rate codes. The thesis work also explores a method of constructing optimal codes by computing the subfield subcodes of Generalized Toric (GT) codes that is a natural extension of RS codes over several dimensions. The polynomial generators or evaluation polynomials for subfield-subcodes of GT codes are identified based on which dimension and bound for the minimum distance are computed. The algebraic structure for the polynomials evaluating to subfield is used to simplify the list decoding algorithm for BCH codes. Finally, an efficient and novel approach is proposed for exploiting powerful codes having complex decoding but simple encoding scheme (comparable to RS codes) for multihop wireless sensor network (WSN) applications

Successive DF relaying: MS-DIS aided interference suppression and three-stage concatenated architecture design

Conventional single-relay aided two-phase cooperative networks employing coherent detection algorithms incur a significant 50% throughput loss. Furthermore, it is unrealistic to expect that in addition to the task of relaying, the relay-station would dedicate further precious resources to the estimation of the source-relay channel in support of coherent detection. In order to circumvent these problems, we propose decode and-forward (DF) based successive relaying employing noncoherent detection schemes. A crucial challenge in this context is that of suppressing the successive relaying induced interference, despite dispensing with any channel state information (CSI). We overcome this challenge by introducing a novel adaptive Newton algorithm based multiple-symbol differential interference suppression (MS-DIS) scheme. Correspondingly, a three-stage concatenated transceiver architecture is devised. We demonstrate that our proposed system is capable of near-error-free transmissions at low signal-to-noise ratios

Lowering qubit requirements for quantum simulations of fermionic systems

The mapping of fermionic states onto qubit states, as well as the mapping of fermionic Hamiltonian into quantum gates enables us to simulate electronic systems with a quantum computer. Benefiting the understanding of many-body systems in chemistry and physics, quantum simulation is one of the great promises of the coming age of quantum computers. One challenge in realizing simulations on near-term quantum devices is the large number of qubits required by such mappings. In this work, we develop methods that allow us to trade-off qubit requirements against the complexity of the resulting quantum circuit. We first show that any classical code used to map the state of a fermionic Fock space to qubits gives rise to a mapping of fermionic models to quantum gates. As an illustrative example, we present a mapping based on a non-linear classical error correcting code, which leads to significant qubit savings albeit at the expense of additional quantum gates. We proceed to use this framework to present a number of simpler mappings that lead to qubit savings with only a very modest increase in gate difficulty. We discuss the role of symmetries such as particle conservation, and savings that could be obtained if an experimental platform could easily realize multi-controlled gates.Comment: 11+13 pages, 5 figures, 2 tables, see ArXiv files for Mathematica code (text file) and documentation (pdf); fixed typos in this new versio

Successive-relaying-aided decode-and-forward coherent versus noncoherent cooperative multicarrier space–time shift keying

Abstract—Successive-relaying-aided (SR) cooperative multi-carrier (MC) space–time shift keying (STSK) is proposed for frequency-selective channels. We invoke SR to mitigate the typical 50% throughput loss of conventional half-duplex relaying schemes and MC code-division multiple access (MC-CDMA) to circumvent the dispersive effects of wireless channels and to reduce the SR-induced interference. The distributed relay terminals form two virtual antenna arrays (VAAs), and the source node (SN) successively transmits frequency-domain (FD) spread signals to one of the VAAs, in addition to directly transmitting to the destination node (DN). The constituent relay nodes (RNs) of each VAA activate cyclic-redundancy-checking-based (CRC) selective decode-and-forward (DF) relaying. The DN can jointly detect the signals received via the SN-to-DN and VAA-to-DN links using a low-complexity single-stream-based joint maximum-likelihood (ML) detector. We also propose a differentially encoded cooperative MC-CDMA STSK scheme to facilitate communications over hostile dispersive channels without requiring channel estimation (CE). Dispensing with CE is important since the relays cannot be expected to altruistically estimate the SN-to-RN links for simply supporting the source. Furthermore, we propose soft-decision-aided serially concatenated recursive systematic convolutional (RSC) and unity-rate-coded (URC) cooperative MC STSK and investigate its performance in both coherent and noncoherent scenarios

On connectivity-dependent resource requirements for digital quantum simulation of dd-level particles

A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-$s$, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by $d$-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of SWAP gates required to Trotterize commonly used quantum operators, we use both analytical arguments and automatic tools that optimize the schedule in multiple stages. We study the unary (or one-hot), Gray, standard binary, and block unary encodings, with three connectivities: linear array, ladder array, and square grid. Among other trends, we find that while the ladder array leads to substantial efficiencies over the linear array, the advantage of the square over the ladder array is less pronounced. These results are applicable in hardware co-design and in choosing efficient qudit encodings for a given set of near-term quantum hardware. Additionally, this work may be relevant to the scheduling of other quantum algorithms for which matrix exponentiation is a subroutine.Comment: Accepted to QCE20 (IEEE Quantum Week). Corrected erroneous circuits in Figure