Attaining Capacity with iterated (U∣U+V)(U |U + V ) codes based on AG codes and Koetter-Vardy soft decoding

International audienceIn this paper we show how to attain the capacity of discrete symmetric channels with polynomial time decoding complexity by considering iterated (U | U + V) constructions with algebraic geometry (AG) code components. These codes are decoded with a recursive computation of the a posteriori probabilities of the code symbols together with decoding the AG components with the Koetter-Vardy algorithm. We show that, when the number of levels of the iterated (U | U + V) construction tends to infinity, we attain the capacity of any discrete symmetric channel. Moreover the error probability decays quasi-exponentially with the codelength in the case of Reed-Solomon code constituents and exponentially with Tsfasman-Vl˘ aduts-Zink code constituents

Self-concatenated coding for wireless communication systems

In this thesis, we have explored self-concatenated coding schemes that are designed for transmission over Additive White Gaussian Noise (AWGN) and uncorrelated Rayleigh fading channels. We designed both the symbol-based Self-ConcatenatedCodes considered using Trellis Coded Modulation (SECTCM) and bit-based Self- Concatenated Convolutional Codes (SECCC) using a Recursive Systematic Convolutional (RSC) encoder as constituent codes, respectively. The design of these codes was carried out with the aid of Extrinsic Information Transfer (EXIT) charts. The EXIT chart based design has been found an efficient tool in finding the decoding convergence threshold of the constituent codes. Additionally, in order to recover the information loss imposed by employing binary rather than non-binary schemes, a soft decision demapper was introduced in order to exchange extrinsic information withthe SECCC decoder. To analyse this information exchange 3D-EXIT chart analysis was invoked for visualizing the extrinsic information exchange between the proposed Iteratively Decoding aided SECCC and soft-decision demapper (SECCC-ID). Some of the proposed SECTCM, SECCC and SECCC-ID schemes perform within about 1 dB from the AWGN and Rayleigh fading channels’ capacity. A union bound analysis of SECCC codes was carried out to find the corresponding Bit Error Ratio (BER) floors. The union bound of SECCCs was derived for communications over both AWGN and uncorrelated Rayleigh fading channels, based on a novel interleaver concept.Application of SECCCs in both UltraWideBand (UWB) and state-of-the-art video-telephone schemes demonstrated its practical benefits.In order to further exploit the benefits of the low complexity design offered by SECCCs we explored their application in a distributed coding scheme designed for cooperative communications, where iterative detection is employed by exchanging extrinsic information between the decoders of SECCC and RSC at the destination. In the first transmission period of cooperation, the relay receives the potentially erroneous data and attempts to recover the information. The recovered information is then re-encoded at the relay using an RSC encoder. In the second transmission period this information is then retransmitted to the destination. The resultant symbols transmitted from the source and relay nodes can be viewed as the coded symbols of a three-component parallel-concatenated encoder. At the destination a Distributed Binary Self-Concatenated Coding scheme using Iterative Decoding (DSECCC-ID) was employed, where the two decoders (SECCC and RSC) exchange their extrinsic information. It was shown that the DSECCC-ID is a low-complexity scheme, yet capable of approaching the Discrete-input Continuous-output Memoryless Channels’s (DCMC) capacity.Finally, we considered coding schemes designed for two nodes communicating with each other with the aid of a relay node, where the relay receives information from the two nodes in the first transmission period. At the relay node we combine a powerful Superposition Coding (SPC) scheme with SECCC. It is assumed that decoding errors may be encountered at the relay node. The relay node then broadcasts this information in the second transmission period after re-encoding it, again, using a SECCC encoder. At the destination, the amalgamated block of Successive Interference Cancellation (SIC) scheme combined with SECCC then detects and decodes the signal either with or without the aid of a priori information. Our simulation results demonstrate that the proposed scheme is capable of reliably operating at a low BER for transmission over both AWGN and uncorrelated Rayleigh fading channels. We compare the proposed scheme’s performance to a direct transmission link between the two sources having the same throughput

A STUDY OF LINEAR ERROR CORRECTING CODES

Since Shannon's ground-breaking work in 1948, there have been two main development streams of channel coding in approaching the limit of communication channels, namely classical coding theory which aims at designing codes with large minimum Hamming distance and probabilistic coding which places the emphasis on low complexity probabilistic decoding using long codes built from simple constituent codes. This work presents some further investigations in these two channel coding development streams. Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents and Mattson-Solomon polynomials, and are complementary to each other. The two methods generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and projective geometry codes. Their extension to non binary fields is shown to be straightforward. These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative decoding. It is also shown that for some of these codes, maximum likelihood performance may be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords of the dual code for each iteration. Following a property of the revolving-door combination generator, multi-threaded minimum Hamming distance computation algorithms are developed. Using these algorithms, the previously unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated. In addition, the highest minimum Hamming distance attainable by all binary cyclic codes of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes which have higher minimum Hamming distance than the previously considered best known linear code have been found. It is shown that by exploiting the structure of circulant matrices, the number of codewords required, to compute the minimum Hamming distance and the number of codewords of a given Hamming weight of binary double-circulant codes based on primes, may be reduced. A means of independently verifying the exhaustively computed number of codewords of a given Hamming weight of these double-circulant codes is developed and in coiyunction with this, it is proved that some published results are incorrect and the correct weight spectra are presented. Moreover, it is shown that it is possible to estimate the minimum Hamming distance of this family of prime-based double-circulant codes. It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection mechanism that offers much better throughput and performance than the conventional ORG scheme is described. Using the same method it is shown that the performance of conventional CRC scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy communications system and it is shown that sequences of good error correction codes, suitable for use in incremental redundancy communications systems may be obtained using the Constructions X and XX. Examples are given and their performances presented in comparison to conventional CRC schemes

Near-capacity fixed-rate and rateless channel code constructions

Fixed-rate and rateless channel code constructions are designed for satisfying conflicting design tradeoffs, leading to codes that benefit from practical implementations, whilst offering a good bit error ratio (BER) and block error ratio (BLER) performance. More explicitly, two novel low-density parity-check code (LDPC) constructions are proposed; the first construction constitutes a family of quasi-cyclic protograph LDPC codes, which has a Vandermonde-like parity-check matrix (PCM). The second construction constitutes a specific class of protograph LDPC codes, which are termed as multilevel structured (MLS) LDPC codes. These codes possess a PCM construction that allows the coexistence of both pseudo-randomness as well as a structure requiring a reduced memory. More importantly, it is also demonstrated that these benefits accrue without any compromise in the attainable BER/BLER performance. We also present the novel concept of separating multiple users by means of user-specific channel codes, which is referred to as channel code division multiple access (CCDMA), and provide an example based on MLS LDPC codes. In particular, we circumvent the difficulty of having potentially high memory requirements, while ensuring that each user’s bits in the CCDMA system are equally protected. With regards to rateless channel coding, we propose a novel family of codes, which we refer to as reconfigurable rateless codes, that are capable of not only varying their code-rate but also to adaptively modify their encoding/decoding strategy according to the near-instantaneous channel conditions. We demonstrate that the proposed reconfigurable rateless codes are capable of shaping their own degree distribution according to the nearinstantaneous requirements imposed by the channel, but without any explicit channel knowledge at the transmitter. Additionally, a generalised transmit preprocessing aided closed-loop downlink multiple-input multiple-output (MIMO) system is presented, in which both the channel coding components as well as the linear transmit precoder exploit the knowledge of the channel state information (CSI). More explicitly, we embed a rateless code in a MIMO transmit preprocessing scheme, in order to attain near-capacity performance across a wide range of channel signal-to-ratios (SNRs), rather than only at a specific SNR. The performance of our scheme is further enhanced with the aid of a technique, referred to as pilot symbol assisted rateless (PSAR) coding, whereby a predetermined fraction of pilot bits is appropriately interspersed with the original information bits at the channel coding stage, instead of multiplexing pilots at the modulation stage, as in classic pilot symbol assisted modulation (PSAM). We subsequently demonstrate that the PSAR code-aided transmit preprocessing scheme succeeds in gleaning more information from the inserted pilots than the classic PSAM technique, because the pilot bits are not only useful for sounding the channel at the receiver but also beneficial for significantly reducing the computational complexity of the rateless channel decoder

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum