Elias Gamma Error Correction Code

Abstract

Shannon’s source-channel coding separation theorem states that near-capacity communication is theoretically possible, when employing Separate Source and Channel Codes (SSCCs), provided that an unlimited encoding/decoding delay and complexity can be afforded. However, it is typically impossible to remove all source redundancy with the aid of practical finite-delay and finite-complexity source encoding, which leads to capacity loss. As a potential remedy, Joint Source and Channel Codes (JSCCs) have been proposed for exploiting the residual redundancy and hence for avoiding any capacity loss. However, all previous JSCCs have been designed for representing symbols values that are selected from a set having a low cardinality and hence they suffer from an excessive decoding complexity, when the cardinality of the symbol value set is large, leading to an infinite complexity, when the cardinality is infinite.

Motivated by this, we propose the family of Unary Error Correction (UEC), Elias Gamma Error Correction (EGEC) and Reordered Elias Gamma Error Correction (REGEC) codes in this thesis. Our family of codes belong to the JSCC class designed to have only a modest complexity that is independent of the cardinality of the symbol value set. We exemplify the application of each of the codes in the context of a serially concatenated iterative decoding scheme. In each coding scheme, the encoder generates a bit sequence by encoding and concatenating codewords, while the decoder performs iterative decoding using the classic Logarithmic Bahl, Cocke, Jelinek and Raviv (Log-BCJR) algorithm. Owing to this, our proposed codes are capable of mitigating any potential capacity loss, hence facilitating near-capacity operation.

Our proposed UEC code is the first JSCC that maintains a low decoding complexity, when invoked for representing symbol values that are selected from a set having large or even infinite cardinality. The UEC trellis is designed to describe the unary codewords so that the transitions between its states are synchronous with the transitions between the consecutive codewords in the bit sequence. The unary code employed in the UEC code has a simple structure, which can be readily exploited for error correction without requiring an excessive number of trellis transitions and states. However, the UEC scheme has found limited applications, since the unary code is not a universal code. This motivates the design of our EGEC code, which is the first universal code in our code family. The EGEC code relies on trellis representation of the EG code, which is generated by decomposing each symbol into two sub-symbols, for the sake of simplifying the structure of the EG code. However, the reliance on these two parts requires us to carefully tailor the Unequal Protection (UEP) of the two parts for the specific source probability distribution encountered, whilst the actual source distribution may be unknown or non-stationary. Additionally, the complex structure of the EGEC code may impose further disadvantages associated with an increased decoding delay, loss of synchronisation, capacity loss and increased complexity due to puncturing. This motivates us to propose a universal JSCC REGEC code, which has a significantly simpler structure than the EGEC code. The proposed codes were benchmarked against SSCC benchmarkers throughout this thesis and they were found to offer significant gains in all cases.

Finally, we demonstrate that our code family proposed in this thesis can be extended by several potential directions. The sophisticated techniques that have been subsequently proposed in the thesis for extending the UEC code, such as irregular trellis designs and the adaptive distribution-learning algorithm, can be readily applied to the REGEC codes which is an explicit benefit of its simple trellis structure. Furthermore, our proposed REGEC code can be extended using techniques that been subsequently proposed for extending the EGEC both to Rice Error Correction (RiceEC) codes and to Exponential Golomb Error Correction (ExpGEC) codes.

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Identifiers

ORCID for Lajos Hanzo: orcid.org/0000-0002-2636-5214

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