Publicação
Constructions of optimal convolutional codes
| Resumo: | One of the fundamental challenges in coding theory is the construction of optimal codes that maximize error detection and correction capabilities. In this thesis, we constructed new optimal convolutional codes, focusing on maximum distance separable (MDS) and maximum distance profile (MDP) codes, using algebraic techniques, including matrix structures such as superregular matrices, and novel repetition patterns for generator matrices. We established new criteria for MDS convolutional codes with optimal column distances. We developed an enhanced decoding algorithm for a specific class of optimal binary convolutional codes by integrating Viterbi algorithm features with a generalized Reed-Muller decoding approach, reducing computational complexity. We extended our analysis to multidimensional (mD) convolutional codes, deriving new distance bounds and providing explicit constructions for 3D convolutional codes that meet these bounds. We introduced Pseudo-MDP convolutional codes to improve burst error correction without increasing field size. Our findings support efficient error correction in low-delay communication systems, erasure channels, and multidimensional data transmission, laying the groundwork for further research on small-field convolutional codes, advanced decoding methods, and multidimensional coding applications. |
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| Autores principais: | Abreu, Maria Zita Fiqueli de |
| Assunto: | Convolutional codes Free distance Column distances Maximum distance separable (MDS) codes Maximum distance profile (MDP) codes Decoding algorithms Viterbi algorithm Erasure correction Multidimensional codes |
| Ano: | 2025 |
| País: | Portugal |
| Tipo de documento: | tese de doutoramento |
| Tipo de acesso: | acesso embargado |
| Instituição associada: | Universidade de Aveiro |
| Idioma: | inglês |
| Origem: | RIA - Repositório Institucional da Universidade de Aveiro |
| Resumo: | One of the fundamental challenges in coding theory is the construction of optimal codes that maximize error detection and correction capabilities. In this thesis, we constructed new optimal convolutional codes, focusing on maximum distance separable (MDS) and maximum distance profile (MDP) codes, using algebraic techniques, including matrix structures such as superregular matrices, and novel repetition patterns for generator matrices. We established new criteria for MDS convolutional codes with optimal column distances. We developed an enhanced decoding algorithm for a specific class of optimal binary convolutional codes by integrating Viterbi algorithm features with a generalized Reed-Muller decoding approach, reducing computational complexity. We extended our analysis to multidimensional (mD) convolutional codes, deriving new distance bounds and providing explicit constructions for 3D convolutional codes that meet these bounds. We introduced Pseudo-MDP convolutional codes to improve burst error correction without increasing field size. Our findings support efficient error correction in low-delay communication systems, erasure channels, and multidimensional data transmission, laying the groundwork for further research on small-field convolutional codes, advanced decoding methods, and multidimensional coding applications. |
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