Trellis Structure of Block Codes

We propose an information-theoretic private information retrieval (PIR) scheme for distributed storage systems where data is stored using a linear systematic code of rate R> 1/2. The proposed scheme generalizes the PIR scheme for data stored using maximum distance separable codes recently proposed by Tajeddine and El Rouayheb for the scenario of a single spy node. We further propose an algorithm to optimize the communication price of privacy (cPoP) using the structure of the underlying linear code. As an example, we apply the proposed algorithm to several distributed storage codes, showing that the cPoP can be significantly reduced by exploiting the structure of the distributed storage code.

We establish a general theorem which, given an arbitrary linear code, generates a family of linear codes of larger lengths and higher dimensions. The proof is based on a representation of the given linear code as a (generalized) algebraic-geometry code. Examples yielding many optimal linear codes document the power of this theorem.

In this work, the homomorphism of the classic linear block code in linear network coding for the case of binary field and its extensions is studied. It is proved that the classic linear error-control block code is homomorphic network error-control code in network coding. That is, if the source packets at the source node for a linear network coding are precoded using a linear block code, then every packet flowing in the network regarding to the source satisfies the same constraints as the source. As a consequence, error detection and correction can be performed at every intermediate nodes of multicast flow, rather than only at the destination node in the conventional way, which can help to identify and correct errors timely at the error-corrupted link and save the cost of forwarding error-corrupted data to the destination node when the intermediate nodes are ignorant of the errors. In addition, three examples are demonstrated which show that homomorphic linear code can be combined with homomorphic signature, McEliece public-key cryptosystem and unequal error protection respectively and thus have a great potential of practical utility.

Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.

Linear block codes are studied for improving the reliability of message storage in computer memory with stuck-at defects and noise. The case when the side information about the state of the defects is available to the decoder or to the encoder is considered. In the former case, stuck-at cells act as erasures so that techniques for decoding linear block codes for erasures and errors can be directly applied. We concentrate on the complimentary problem of incorporating stuck-at information in the encoding of linear block codes. An algebraic model for stuck-at defects and additive errors is presented. The notion of a "partitioned" linear block code is introduced to mask defects known at the encoder and to correct random errors at the decoder. The defect and error correction capability of partitioned linear block codes is characterized in terms of minimum distances. A class of partitioned cyclic codes is introduced. A BCH-type bound for these cyclic codes is derived and employed to construct partitioned linear block codes with specified bounds on the minimum distances. Finally, a probabilistic model for the generation of stuck-at cells is presented. It is shown that partitioned linear block codes achieve the Shannon capacity for a computer memory with symmetric defects and errors.

Iterative soft decision decoding of linear block codes is a practical necessity when working with even modest block lengths. A number of algorithms have been proposed in the literature which use the permutation group of a code and the belief propagation (BP) algorithm for decoding. A novel soft-input, soft-output algorithm is presented that can be used for efficiently decoding of linear cyclic codes. Utilising the automorphism property of cyclic codes the permutation is incorporated into the belief propagation algorithm resulting in faster convergence and better error correcting performance. Performance of the new approach is analysed using a (63,45) BCH code and a (72,36) quadratic residue code. Simulation results show significant reduction in the average number of required decoding iterations and some improvement in error correcting performance over published algorithms.

Orthogonal Frequency Division Multiplexing (OFDM) is used to encode digital data at multiple carrier frequencies and has evolved into a very common broadband digital communication scheme. All its subcarriers are orthogonal to each other, reducing the interference among subcarriers. Additionally, inter-symbol interference (ISI) can be reduced by using the Cyclic Prefix (CP). However, the intra-block OFDM symbol still degrades the performance. Bit Error Rate (BER) can be used to analyze OFDM performance. The efficiency of the OFDM system increases as the BER value decreases. The performance of the errors can be enhanced by using forward error correction. This study covers the performance of OFDM with Linear Block Code, Convolutional Code, Cyclic Block Code, Turbo Code, and LDPC across the whole SNR range. There has also been a discussion of the difference between coded and uncoded OFDM. The outcome demonstrates that Cyclic Block Code is superior to Linear Block Code in complex situations., LDPC Code performs better when increasing the data rate than Convolutional Code, and with more code generator polynomials, Turbo Code shows less BER value in comparison with Convolutional Code.

In this chapter, we discuss a number of codes for error control. Only block codes are treated here. Discussion on convolutional codes will be deferred until next chapter. After reviewing some information theoretic foundations of coding in the first section, linear block codes are treated in Section 3.2. The concepts of parity-check and generator matrices to represent linear block codes are discussed. Several examples of block codes are given, including the important class of Hamming codes. Principles behind syndrome decoding and decoding using a standard array are treated in Section 3.3. Section 3.4 provides some useful bounds on coding and introduces the concept of coding gain. Section 3.5 discusses the principles behind cyclic codes. Some important decoding techniques for these codes are treated in Section 3.6. These include the Meggitt and error-trapping decoders. After introducing some algebra in Section 3.7, in the next three sections that follow, we treat the most important and practical of all cyclic codes, the Bose-Chaudhuri-Hocquenghem (BCH) codes and Reed-Solomon codes. The treatment includes the MasseyBerlekamp algorithm for decoding these codes. In Section 3.11, we turn to coding for burst error control, which has been successfully applied to storage media such as magnetic tapes and compact disc. Automatic-repeat-request (ARQ) schemes find wide applicability in computer networks and these schemes are treated in the last section.

Chapter 6 - Information Theory and Classical Error Correcting Codes

Decoding a redundant code, when redefined as a means for estimating a posterior probability distribution given a prior one, has a simple output if one looks for the posterior distribution in a form separable with respect to the information symbols. The optimal distribution of this form results from minimizing its cross-entropy with respect to the prior distribution, which for a linear code leads to a system of implicit analog equations to be solved. This can be done by iteration, but the system has multiple solutions so that determining proper initial values becomes the main difficulty. We interpret any linear code as a kind of product of its parity checks whose successive decodings provide adequate initial values for solving the system, provided the received symbols are reordered by decreasing reliability (except for few inversions in order to keep the generator matrix in systematic form). Simulation results are presented.

Future broadband wireless communication systems will be characterized by heterogeneity with regard to node capabilities, link level requirements, cell structures etc. High carrier frequencies and large system bandwidth will make extreme diversity factors feasible. We consider a recently proposed class of linear space-time block codes (and associated decoders), which meet the requirements of future communication systems. In the case of these codes an increase of the block length can lead to higher diversity factors but the required decoder complexity increases too. It is not always possible for the transmitter to adapt the used block length to the decoder complexity of every receiver (e.g. for a broadcast transmission or if the transmitter has no knowledge about the receiver complexity), especially when high diversity gains are favorable. In this paper we present an approach to low complexity decoding by subspace partitioning. So, if a transmitter uses large block lengths, low complexity nodes can handle these blocks and achieve moderate diversity gains while nodes with higher capabilities can profit from high diversity gains.

The problem of decoding an (n, k, d) error-correcting block code, where a k-bit message word is mapped to an n-bit codeword, can be cast as a single-label classification problem. While it has been observed that the performance of such single-label neural decoders closely approaches that of the corresponding maximum likelihood decoder (MLD), the number of output nodes increases exponentially with k, making it prohibitive to implement for large k. To address this issue, we explore classification based multi-label neural decoders, in which the number of output nodes increases linearly with k. We consider well-known linear and non-linear block codes, as well as concatenated block codes, which have applications in emerging wireless networks. Our study finds that (i) although the number of output nodes linearly increases with k in a multi-label decoder, it requires more hidden layers and nodes in each hidden layer than the corresponding single-label decoder to achieve its best performance, and (ii) although one can design a multi-label decoder with bit error rate matching that of the MLD, it leaves more blocks in error leading to a reduced performance in terms of block error rate. We also note that the performance of the proposed decoder for concatenated codes is at least as good as that of a natural decoding algorithm in which the inner code is first decoded using the MLD and then the outer code is decoded with a polynomial-time decoding algorithm.

Entanglement-assisted quantum error-correcting codes can be seen as a new-type of quantum error-correcting codes and can be constructed from arbitrary linear codes which should not satisfy the dual-containing condition by utilizing shared entangled states between the sender and the receiver in advance. In this paper, we construct several new classes of entanglement-assisted quantum maximum-distance-separable codes from constacyclic codes and cyclic codes by exploiting small pre-shared entangled states, respectively. These codes are new in the sense that they are not covered by the codes available in the literature.

Entanglement-assisted quantum error-correcting code (EAQECC) is a new-type quantum code and can be constructed from arbitrary linear codes by sharing entangled states c in advance. In this letter, we explore a family of entanglement-assisted quantum error-correcting codes (EAQECCs) with length (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - 1)/12 from negacyclic codes. Compared with known results in the literatures, our EAQECCs are new and have larger minimum distance.

We propose a private information retrieval (PIR) protocol for distributed storage systems (DSSs) with noncolluding nodes where data is stored using an arbitrary linear code. An expression for the PIR rate, i.e., the ratio of the amount of retrieved stored data per unit of downloaded data, is derived, and a necessary and a sufficient condition for codes to achieve the PIR capacity are given. The necessary condition is based on the generalized Hamming weights of the storage code, while the sufficient condition is based on code automorphisms. We show that cyclic codes and Reed-Muller codes satisfy the sufficient condition and are thus PIR capacity-achieving.