QPSK Block-Modulation Codes for Unequal Error Protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated

Moment balancing templates: constructions to add insertion/deletion correction capability to error correcting or constrained codes

Abstract: Templates are constructed to extend arbitrary additive error correcting or constrained codes, i.e., additional redundant bits are added in selected positions to balance the moment of the codeword. The original codes may have error correcting capabilities or constrained output symbols as predetermined by the usual communication system considerations, which are retained after extending the code. Using some number theoretic constructions in the literature, insertion/deletion correction can then be achieved. If the template is carefully designed, the number of additional redundant bits for the insertion/deletion correction can be kept small—in some cases of the same order as the number of parity bits in a Hamming code of comparable length

Error-corrected quantum repeaters with GKP qudits

The Gottesman-Kitaev-Preskill (GKP) code offers the possibility to encode higher-dimensional qudits into individual bosonic modes with, for instance, photonic excitations. Since photons enable the reliable transmission of quantum information over long distances and since GKP states subject to photon loss can be recovered to some extent, the GKP code has found recent applications in theoretical investigations of quantum communication protocols. While previous studies have primarily focused on GKP qubits, the possible practical benefits of higher-dimensional GKP qudits are hitherto widely unexplored. In this paper, we carry out performance analyses for three quantum repeater protocols based on GKP qudits including concatenations with a multi-qudit quantum polynomial code. We find that the potential data transmission gains for qudits are often hampered by their decreased GKP error-correcting capabilities. However, we also identify parameter regimes in which having access to an increased number of quantum levels per mode can enhance the theoretically achievable secret-key rate of the quantum repeater. Some of our protocols share the attractive feature that local processing and complete error syndrome identification are realizable without online squeezing. Provided a supply of suitable multi-mode GKP states is available, this can be realized with a minimal set of passive linear optical operations, even when the logical qudits are composed of many physical qudits.Comment: 19+11 pages, 6+4 figures. Comments welcom

Construction of lattices for communications and security

In this thesis, we propose a new class of lattices based on polar codes, namely polar lattices. Polar lattices enjoy explicit construction and provable goodness for the additive white Gaussian noise (AWGN) channel, \textit{i.e.}, they are \emph{AWGN-good} lattices, in the sense that the error probability (for infinite lattice coding) vanishes for any fixed volume-to-noise ratio (VNR) greater than $2\pi e$. Our construction is based on the multilevel approach of Forney \textit{et al.}, where on each level we construct a capacity-achieving polar code. We show the component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We present a more precise analysis of the VNR of the resultant lattice, which is upper-bounded in terms of the flatness factor and the capacity losses of the component codes. The proposed polar lattices are efficiently decodable by using multi-stage decoding. Design examples are presented to demonstrate the superior performance of polar lattices. However, there is no infinite lattice coding in the practical applications. We need to apply the power constraint on the polar lattices which generates the polar lattice codes. We prove polar lattice codes can achieve the capacity \frac{1}{2}\log(1+\SNR) of the power-constrained AWGN channel with a novel shaping scheme. The main idea is that by implementing the lattice Gaussian distribution over the AWGN-good polar lattices, the maximum error-free transmission rate of the resultant coding scheme can be arbitrarily close to the capacity \frac{1}{2}\log(1+\SNR). The shaping technique is based on discrete lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. Then it is straightforward to employ multilevel asymmetric polar codes which is a combination of polar lossless source coding and polar channel coding. The construction of polar codes for an asymmetric channel can be converted to that for a related symmetric channel, and it turns out that this symmetric channel is equivalent to an minimum mean-square error (MMSE) scaled $\Lambda/\Lambda'$ channel in lattice coding in terms of polarization, which eventually simplifies our coding design. Finally, we investigate the application of polar lattices in physical layer security. Polar lattice codes are proved to be able to achieve the strong secrecy capacity of the Mod-$\Lambda$ AWGN wiretap channel. The Mod-$\Lambda$ assumption was due to the fact that a practical shaping scheme aiming to achieve the optimum shaping gain was missing. In this thesis, we use our shaping scheme and extend polar lattice coding to the Gaussian wiretap channel. By employing the polar coding technique for asymmetric channels, we manage to construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. Then we prove the resultant wiretap coding scheme can achieve the strong secrecy capacity for the Gaussian wiretap channel.Open Acces