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Hyperentanglement, the entangled states in the multiple degrees of freedom (DOFs) of a photon system simultaneously, has attracted much attention recently as it can increase the capacity of quantum communication largely. Here we present the first scheme for concentrating the nonlocal two-photon six-qubit systems entangled in polarization, spatial-mode, and time-bin DOFs with linear optics. We construct the parity-check gates for the quantum states of nonlocal photon systems in all the three DOFs with linear-optical elements. Fortunately, our parity-check gate for one DOF does not affect the quantum states in the other two DOFs, which makes our hyperentanglement concentration protocol (hyper-ECP) can be implemented conveniently for polarization–spatial–time-bin partially hyperentangled states. This hyper-ECP is practical as it requires only linear-optical elements, and it can be directly generalized to improve the entanglement of nonlocal multi-photon systems entangled in three DOFs in the long-distance high-capacity quantum communication. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes' propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1-2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10-25 iterations less than the standard BP decoder. [ABSTRACT FROM AUTHOR]

This paper presents a compressive tracking framework using distributed binary sensors. The goal of this research is to achieve the minimum data throughput for an accurate multitarget tracking system through novel spatial sampling schemes. The framework consists of two main components: space encoding and measurement decoding. The space encoding scheme is based on the low-density parity-check matrix, which converts $k$ -sparse target position vectors into different codewords. The measurement decoding scheme contains linear-programming-based localization and graphical-model-based tracking algorithms, which converts codewords into the states of multiple targets. A posterior Cramér–Rao bound analysis is utilized to achieve the tradeoff between the compression ratio of measurements and the accuracy of the tracking system. Simulation and experimental results are provided to validate the proposed framework. [ABSTRACT FROM AUTHOR]

We consider the joint detection and decoding of m-sequences. The receiver has to decide whether an m-sequence is received and possibly to decode its initial state. To do so, it implements an iterative message-passing decoding algorithm that operates on a parity check matrix, built upon a number of reference parity-check equations satisfied by the m-sequence. This matrix concatenates several elementary parity check matrices which are derived from reference equations. Unlike the conventional decoding case, the detection problem imposes to consider false alarms that may occur when the decoder is only fed with noise. While absorbing sets are known to be responsible for the error floor phenomenon of iterative message-passing decoders, we show that they may have a beneficial effect on the detection performance, in that they may prevent the decoder to produce false alarms. We further compute the number of hybrid cycles of length six and eight in the Tanner graph of the decoder and use the minimization of this number as criterion to derive an algorithm for selecting the reference parity check equations. This algorithm was found to be efficient for minimizing the probability of false alarm and decreases also the probability of wrong detection in the very small SNR region. This has been achieved at the cost of a reduction of the probability of correct detection. [ABSTRACT FROM PUBLISHER]

Byte correcting perfect codes are developed to correct burst errors within bytes. If a code is byte correcting code and we say that the code is -burst correcting, meaning that it corrects a single burst of length or less within a byte. A byte correcting code is such that if ; denote the set of syndromes obtained from the th-byte of the parity check matrix and ; denote the set of syndromes obtained from the th-byte of the parity check matrix then . In an code, if there are bytes of size , then . Byte correcting codes are preferred where information stored in all bytes are equally important. But there are cases where some parts of the message are more important than other parts of the message, for example, if we have to transmit a message on a border location ' Shift Battalion (Bn) from Location A to Location B' then, we will focus more on the information shift, location and location i.e., bytes containing these information will be more important than others. In this situation, it is needed that during transmission these bytes should have no possibility of error. In other words, these bytes should be protected absolutely against any error. Keeping this in mind, we study burst error correcting capabilities of byte oriented codes in terms of byte protection level of each byte. If there is a byte error pattern of length in the transmission then all those bytes of the received pattern will be decoded correctly whose burst protection level is or more even though the code word may be decoded wrongly. Taking the code length to be divided into bytes with burst protection level of the th-byte as ; ; , we construct linear codes that we call byte protecting burst (BPB) codes and investigate their byte protecting capabilities in this paper. [ABSTRACT FROM AUTHOR]

We propose a new construction for binary locally repairable codes (LRCs) based on a (partial) $t$ -spread, which constructs an LRC with minimum distance $d \geq 6$ and any locality $r \geq 2$ . Furthermore, we found that there are some cases in which the resulting codes are optimum in terms of the upper bound by Cadambe and Mazumdar in 2015. [ABSTRACT FROM PUBLISHER]

Given an n × n matrix A over a field F and a scalar a ∈ F , we consider the linear codes C ( A , a ) : = { B ∈ F n × n | A B = a B A } of length n 2 . We call C ( A , a ) a twisted centralizer code . We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when a = 1 ) is at most n , however for a ≠ 0 , 1 the minimal distance can be much larger, as large as n 2 . [ABSTRACT FROM AUTHOR]

Generalized product codes (GPCs) are extensions of product codes (PCs), where code symbols are protected by two component codes but not necessarily arranged in a rectangular array. We consider a deterministic construction of GPCs (as opposed to randomized code ensembles) and analyze the asymptotic performance over the binary erasure channel under iterative decoding. Our code construction encompasses several classes of GPCs previously proposed in the literature, such as irregular PCs, blockwise braided codes, and staircase codes. It is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity. We show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph. Using a convergence result for these graphs, we derive the density evolution equations that characterize the asymptotic decoding performance. As an application, we discuss the design of irregular GPCs, employing a mixture of component codes with different erasure-correcting capabilities. [ABSTRACT FROM PUBLISHER]

The LDPC codes are Shannon Limit codes that can achieve low bit error rates for SNR applications. The features of LDPC Codes are reduction in the decoding time, latency and as well as no error-floors at high SNRs. The proposed algorithms are SBF, MSA, and MLDD. The various decoding algorithms have been compared for these codes. The parameters are describing the which algorithm further helps in selecting the better decoder used for Medical and Signal Processing Applications. These codes are also used in Generating the Barcodes depends on the size of the Parity Check matrix. [ABSTRACT FROM AUTHOR]

Error correcting codes are required to ensure reliable communication of digitally encoded information. One of the areas of practical importance in which a parallel growth of the subject error correcting codes took place is that of burst error detecting and correcting codes. The nature of burst errors differs from channel to channel depending upon the behavior of channels or the kind of errors which occur during the process of transmission. The rate of transmission is efficient if the number of parity-check digits are as minimum as possible. It is usually not possible to give the exact number of parity-check digits required for a given code. However, bounds can be obtained over the number of parity-check digits. An upper bound for a linear code capable of detecting/correcting burst errors or its variants is many a times established by the technique used to establish Varsharmov-Gilbert-Sacks bound by constructing a parity-check matrix for the requisite code. This technique not only ensures the existence of such a code but also gives a method for constructing such a code. The synthesis method using this technique is cumbersome and to the best of our knowledge, there is no systematic way to construct a paritycheck matrix for a burst error correcting non-binary linear code. Extending the algorithm for binary linear codes given by the authors to non-binary codes, the paper proposes a new algorithm for constructing a parity-check matrix for any linear code over GF(q) capable of detecting and correcting a new kind of burst error called 'm-repeated burst error of length b or less'. Codes based on the proposed algorithm have been illustrated. [ABSTRACT FROM AUTHOR]

Motivated by applications involving hot data, the notion of (r, t) -availability was introduced, where a symbol is said to have (r, t) -availability if it can be reconstructed, respectively, from t disjoint repair alternatives of other symbols, each of size at most r . The key to constructing a family of locally repairable codes with (r, t) -availability is the design of a membership matrix for dividing global parities into local ones such that each repair group contains only one parity symbol. Although explicit designs of membership matrices are available, it remains unclear whether membership matrices with significantly better parameters can be constructed, which was a question left open. To tackle the open problem, this paper first provides a connection between the design of membership matrices and the design of a combinatorial object in the well-known combinatorial design theory, called resolvable configurations. This paper then proposes several direct designs of membership matrices based on Euclidean geometry, finite fields, circulant permutation matrices, affine permutation matrices, and Reed–Solomon codes. By leveraging the notion of resolvable configurations and the Kronecker product, this paper further proposes a two-step design of membership matrices. The proposed direct and two-step designs of membership matrices are shown to have significantly better parameters than those recently available. [ABSTRACT FROM AUTHOR]

The study is devoted to examining the 'efficiency' of Indian foreign exchange (rupee/dollar) market and the relevance of Covered Interest Rate Arbitrage Parity (CIRAP) doctrine therein over the period January 3, 2011 through November 2, 2015. ARIMA (4, 1, 0) stochastic structure of monthly spot rate (st) has been used to generate one-period ahead forecast (set+1) series. These forecasts are Minimum Mean Square Error (MMSE) forecasts and 'rational' by nature. Forward rates (tFt+1) served as the 'unbiased predictor' of the spot rate (st+1) implying that CIRAP did hold well in the market. Again absence of 'risk premium' testifies to the efficiency of the Indian foreign exchange (rupee/dollar) market over the period of study. [ABSTRACT FROM AUTHOR]

In this paper, the belief propagation (BP) based approximation methods for low-density parity-check codes are adapted to the Luby transform soft decoder structure in order to reduce its computational complexity. Moreover, the log-likelihood ratio based adaptive demodulation algorithm is combined with the BP and BP-based algorithms to further reduce the computational complexities and optimum key parameters are determined for normalized min-sum and offset min-sum algorithms. The bit error rate performances of the algorithms over the binary input additive white Gaussian noise channel are obtained by both theoretically and simulations. For theoretical analysis, the Monte-Carlo based density evolution method is used. In addition, computational complexity analyzes of methods are presented. Results show that the computational complexity can be reduced significantly by using combined methods which cost limited signal to noise ratio loss. [ABSTRACT FROM AUTHOR]

Puncturing is one of the methods of increasing the code rate, and the original code before puncturing is called the mother code. Any ( N, K) convolutional code is obtainable by puncturing some ( n,1) mother codes. The objective of a blind recognition of a channel code is to obtain its generator from the intercepted noisy bit stream. The process of the blind recognition of punctured convolutional codes consists of two parts: the reconstruction of the PGM of the ( N, K) punctured convolutional code and the searching process of the mother code and its puncturing pattern. The process of finding the mother code is important for designing the optimum channel decoder. In this paper, a new searching algorithm with the computational complexity of O( K ) polynomial operations is proposed, compared to the existing searching algorithm by M. Cluzeau which requires O( K ) polynomial operations. [ABSTRACT FROM AUTHOR]

Gallager’s decoding algorithms B (GDB) and E (GDE) for low-density parity-check codes have much lower computational complexity and much less required memory size than the sum-product algorithm (SPA). This is because GDB and GDE only use binary or integer operations, while the SPA requires real operations and a look-up table. However, they are hardly used in commercial communication systems since they have a worse performance than the SPA. Layered-division multiplexing (LDM) is considered in ATSC 3.0 in order to deliver multiple broadcasting streams with distinct robustness in a single radio frequency channel. Due to the unique characteristic of the LDM, we propose to use GDB or GDE rather than the SPA for decoding the core layer signal at fixed receivers. Numerical results show that the computational complexity and the required memory size can be reduced without any performance loss by about 50 percent and 80 percent, respectively, when GDB and GDE are employed. [ABSTRACT FROM AUTHOR]

A low-complexity decoding scheme of low-density parity-check codes for the layered-division multiplexing (LDM) systems in ATSC 3.0 is proposed. Based on the unique characteristic of the LDM and the special structure of a parity-check matrix (PCM) for the core layer, the proposed scheme applies the sum-product algorithm (SPA) to only a part of the PCM. For this reason, it is referred to as “the partial SPA,” compared with the conventional scheme, denoted by “the full SPA.” Clearly, it has a significantly lower complexity than the full SPA. Furthermore, we propose hybrid decoders employing both the full SPA and the partial SPA to achieve a better performance, while maintaining their complexity low. Numerical results show that the proposed schemes have a good performance enough to perform successive interference cancellation for the enhanced layer of the LDM systems in ATSC 3.0 as well as they have a low-complexity merit depending on system parameters. Since they only schedule the decoding procedure without modifying the decoder structure, they also have a benefit in the implementation of decoders. [ABSTRACT FROM AUTHOR]

In the orthogonal frequency division multiplexing communications, bit flipping can be used for the reduction of peak-to-average-power ratio (PAPR). Error-correction coding can be used to relieve the degradation of bit-error rates (BERs) resultant from the bit-flipping. We propose designs which appropriately integrate deliberate bit flipping and error-correcting codes with unequal error protection. A modified density evolution method is used for obtaining appropriate code degree distributions to our bit-flipping system. Simulation results show that our proposed designs can obtain effective PAPR reduction with very minor BER degradation in many cases. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the sectionalization in linear tail-biting trellis. In [3], Lafourcade and Vardy investigated the basic property of sectionalization in conventional trellis. Their proofs of basic properties are based on conventional trellis structure which fails to be generalized to linear tail-biting trellis. We derive our proofs from the construction of Koetter-Vardy (KV) trellis in [1]. All properties of sectionalization in conventional trellis are preserved in linear tail-biting trellis. Especially, we prove the necessary condition of an optimal sectionalization in linear tail-biting trellis. [ABSTRACT FROM AUTHOR]

Though the performance of belief propagation (BP) decoder for polar codes is comparable with the successive cancellation (SC) decoder, it has significant performance gap when compared with the improved SC decoders, such as SC list (SCL) decoding. In this paper, we propose an improved BP decoding for polar codes with good performance by adapting their parity-check matrices. The decoding process is iterative and consists of two parts. Firstly, the parity-check matrix of polar codes is adjusted such that one of its submatrices corresponding to less reliable bits is in a sparse nature. Secondly, the BP decoding is applied to the adjusted parity-check matrix. Simulation results show that the proposed decoder, when accompanied with the early termination scheme, provides significant performance gains over the original polar BP decoder under a relatively low decoding complexity and even competes with the cyclic redundancy check (CRC)-aided SCL (CRC-SCL) decoder only by increasing a tolerate complexity. [ABSTRACT FROM AUTHOR]

Generalized integrated interleaved codes refer to two-level Reed-Solomon codes, such that each code of the nested layer belongs to different subcode of the first-layer code. In this paper, we first devise an efficient decoding algorithm by ignoring first-layer miscorrection and by intelligently reusing preceding results during each iteration of a decoding attempt. Neglecting first-layer miscorrection also enables to explicitly and neatly formulate the decoding failure probability. We next derive an erasure correcting algorithm for redundant arrays of independent disks systems. We further construct an algebraic systematic encoding algorithm, which had been open. Analogously, we propose a novel generalized integrated interleaving scheme over binary Bose-Chaudhuri–Hocquenghem codes, reveal a lower bound on the minimum distance, and derive a similar encoding and decoding algorithm as those of Reed-Solomon codes. [ABSTRACT FROM AUTHOR]

The decoding process of a quasi-cyclic low-density parity check (QC-LDPC) code requires a unique type of rotator. These rotators, called multi-size cyclic-shifters (MSCS), rotate the input data with various sizes, where the size means the number of data to be rotated. This paper proposes a low-complexity MSCS structure for the case when the sizes have a non-trivial common divisor. By combining the strong points of two previous structures, the proposed structure achieves the smallest area. The experimental results showed that the area reduction was more than 14.7% when the proposed structure was applied to IEEE 802.16e as an example. [ABSTRACT FROM AUTHOR]

A model describing the dynamics of a foreign exchange (FX) rate should preserve the same level of analytical tractability when the inverted FX process is considered. We show that affine stochastic volatility models satisfy such a requirement. Such a finding allows us to use affine stochastic volatility models as a building block for FX dynamics that are functionally-invariant with respect to the construction of suitable products/ratios of rates, thus generalizing the model of [A. De Col, A. Gnoatto & M. Grasselli (2013) Smiles all around: FX joint calibration in a multi-Heston model, Journal of Banking and Finance 37 (10), 3799-3818.]. [ABSTRACT FROM AUTHOR]

In this paper, we analyze some recently proposed radio-frequency identification group codes based on low-density parity-check matrices that check the integrity of collections of tagged objects. These codes write additional data to the memory of tags to compute the number of missing tags. By studying the underlying requirements for these computations, we obtain bounds and show that there are simpler group codes that perform better, requiring less tag memory. [ABSTRACT FROM AUTHOR]

A fast data synchronization method in optical inter-connected data center is proposed. The origin data are combined to fewer coded data to reduce the synchronization slots. The coded data are broadcast to other nodes. Each node decodes the coded data to recover its missing data. A coding strategy based on coding matrices is proposed for various data distribution. The parity-check matrices in error-control coding theory can be used as coding matrices. The synchronization slots are reduced by the proposed coding method, especially when the missing ratio is low. The slots can be further reduced when the coding field is extended to high-order galois fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new characterization for leafless elementary trapping sets (LETSs) of variable-regular low-density parity-check codes. Recently, Karimi and Banihashemi proposed a characterization of LETSs, which was based on viewing an LETS as a layered superset (LSS) of a short cycle in the code’s Tanner graph. A notable advantage of LSS characterization is that it corresponds to a simple LSS-based search algorithm (expansion technique) that starts from short cycles of the graph and finds the LETSs with LSS structure efficiently. Compared with the LSS-based characterization of Karimi and Banihashemi, which is based on a single LSS expansion technique, the new characterization involves two additional expansion techniques. The introduction of the new techniques mitigates two problems that LSS-based characterization/search suffers from: 1) exhaustiveness: not every LETS structure is an LSS of a cycle and 2) search efficiency: LSS-based search algorithm often requires the enumeration of cycles with length much larger than the girth of the graph, where the multiplicity of such cycles increases rapidly with their length. We prove that using the three expansion techniques, any LETS structure can be obtained starting from a simple cycle, no matter how large the size of the structure a or the number of its unsatisfied check nodes b are, i.e., the characterization is exhaustive. We also demonstrate that for the proposed characterization/search to exhaustively cover all the LETS structures within the (a,b) classes with and b \leq b_{\max } , for any value of a_{\max }$ and b_{\max } , the length of the short cycles required to be enumerated is less than that of the LSS-based characterization/search. We, in fact, show that such a length for the proposed search algorithm is minimal. We also prove that the three expansion techniques, proposed here, are the only expansions needed for characterization of LETS structures starting from simple cycles in the graph, if one requires each and every intermediate sub-structure to be a LETS as well. Extensive simulation results are provided to show that, compared with LSS-based search, significant improvement in search speed and memory requirements can be achieved. [ABSTRACT FROM PUBLISHER]

ABSTRACT In this paper, we propose a new approach to construct a class of check-hybrid generalized low-density parity-check (CH-GLDPC) codes, which are free of small trapping sets. The approach is based on converting some selected check nodes involving a trapping set into super checks corresponding to a 2-error-correcting component code. Specifically, we follow 2 main purposes to construct the check-hybrid codes; first, on the basis of the knowledge of trapping sets of an LDPC code, single parity checks are replaced by super checks to disable the trapping sets. We show that by converting specified single check nodes, denoted as critical checks, to super checks in a trapping set, the parallel bit flipping decoder corrects the errors on a trapping set. The second purpose is to minimize the rate loss through finding the minimum number of such critical checks. We also present an algorithm to find critical checks in a trapping set of a column-weight 3 LDPC code of girth 8 and then provide upper bounds on the minimum number of such critical checks such that the decoder corrects all error patterns on elementary trapping sets. Guaranteed error correction capability of the CH-GLDPC codes is also studied. We show that a CH-GLDPC code in which each variable node is connected to 2 super checks corresponding to a 2-error-correcting component code corrects up to 5 errors. The results are also extended to column-weight 4 LDPC codes of girth 6. Finally, we investigate eliminating of trapping sets of a column-weight 3 LDPC code of girth 8 using the Gallager B decoding algorithm. [ABSTRACT FROM AUTHOR]

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By utilizing the quantum parity-check detectors (PCDs) based on the weak cross-Kerr nonlinearities, we propose a scheme for realizing the remote quantum information concentration (RQIC) with the quantum channel of four-photon bound entangled state. After the PCDs and single-photon operations, quantum information initially distributed in three spatially separated photons is concentrated back to a single photon without performing any global operations. The success probability of the scheme is almost a unit. The necessary single-photon unitary operations corresponding to possible measurement outcomes are given detailedly. [ABSTRACT FROM AUTHOR]

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p -adic integers and integers modulo p e. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme. [ABSTRACT FROM AUTHOR]

We investigate the effect of inserting extra linearity in the Data Encryption Standard (DES) through appropriate nonsingular linear encodings of the output of the individual S-boxes. More specifically, we examine the general situation when the output of each S-box of the DESis precoded separately intoaproperly constructed copy of the inherent even-weight code of length 4. The study is focused on finding multi-round linear characteristics for thus modified DESciphers having maximal effectiveness. Depending on the particular encodings, it turns out that the effectiveness of interest may be larger but in most cases is smaller than that one for the original DESwith the same number of rounds. The latter means that the complexity of successful linear cryptanalysis against these ciphers will mainly increase comparing to the DESitself. The present research extends inanatural way our previous work[Linear Cryptanalysis and Modified DESwith Parity Check in the S-boxes, LNCS 9540 (2016), pp. 60-78]. [ABSTRACT FROM AUTHOR]

We give a new thought for constructing long-range quantum correlation in quantum many-body systems. Our proposed composite parity-conserved matrix product state has long-range quantum correlation only for two spin blocks where their spin-block length larger than 1 compared to any subsystem only having short-range quantum correlation, and we investigate quantum correlation properties of two spin blocks varying with environment parameter and spacing spin number. We also find that the geometry quantum discords of two nearest-neighbor spin blocks and two next-nearest-neighbor spin blocks become smaller and for other conditions the geometry quantum discord becomes larger than that in any subcomponent, i.e., the increase or the production of the long-range quantum correlation is at the cost of reducing the short-range quantum correlation compared to the corresponding classical correlation and total correlation having no any characteristic of regulation. For nearest-neighbor and next-nearest-neighbor all the correlations take their maximal values at the same points, while for other conditions no whether for spacing same spin number or for different spacing spin numbers all the correlations taking their maximal values are respectively at different points which are very close. We believe that our work is helpful to comprehensively and deeply understand the organization and structure of quantum correlation especially for long-range quantum correlation of quantum many-body systems; and further helpful for the classification, the depiction and the measure of quantum correlation of quantum many-body systems. [ABSTRACT FROM AUTHOR]

Transmission channels underlying modern dense storage systems, e.g., Flash memory and magnetic recording (MR) systems, significantly differ from canonical channels, like additive white Gaussian noise (AWGN) channels. While existing low-density parity-check (LDPC) codes optimized for symmetric, AWGN-like channels are being actively considered for Flash applications, we demonstrate that, due to channel asymmetry, such approaches are inadequate. We introduce a refined definition of absorbing sets, which we call general absorbing sets of type two (GASTs), and study the combinatorial properties of GASTs. We then present the weight consistency matrix (WCM), which succinctly captures key properties in a GAST. Furthermore, we show how to customize the WCM definition such that it suits other special subclasses of GASTs. Based on these new concepts, we then develop a new, general combinatorial code optimization framework, which we call the WCM framework, and demonstrate its effectiveness on the realistic highly-asymmetric normal-Laplace mixture (NLM) Flash channel. Moreover, we show that our framework can be customized to optimize non-binary LDPC (NB-LDPC) codes for other asymmetric channels, channels with memory (incorporated in MR systems), and canonical symmetric channels. For all the channels we have simulated NB-LDPC codes over, the codes optimized using the WCM framework enjoy at least 1 order, and up to nearly 2 orders of magnitude performance gain in the uncorrectable bit error rate (UBER) or the frame error rate (FER) relative to the unoptimized codes. Our simulations also show that codes optimized for symmetric channels are not the best choice for asymmetric channels. [ABSTRACT FROM PUBLISHER]