This thesis presents the results of an investigation into the use of puzzle-based combinatorial structures for erasure correction purposes. The research encompasses two main combinatorial structures: the well-known number placement puzzle Sudoku and a novel three component construction designed specifically with puzzle-based erasure correction in mind. The thesis describes the construction of outline erasure correction schemes incorporating each of the two structures. The research identifies that both of the structures contain a number of smaller sub-structures, the removal of which results in a grid with more than one potential solution - a detrimental property for erasure correction purposes. Extensive investigation into the properties of these sub-structures is carried out for each of the two outline erasure correction schemes, and results are determined that indicate that, although the schemes are theoretically feasible, the prevalence of sub-structures results in practically infeasible schemes. The thesis presents detailed classifications for the different cases of sub-structures observed in each of the outline erasure correction schemes. The anticipated similarities in the sub-structures of Sudoku and sub-structures of Latin Squares, an established area of combinatorial research, are observed and investigated, the proportion of Sudoku puzzles free of small sub-structures is calculated and a simulation comparing the recovery rates of small sub-structure free Sudoku and standard Sudoku is carried out. The analysis of sub-structures for the second erasure correction scheme involves detailed classification of a variety of small sub-structures; the thesis also derives probabilistic lower bounds for the expected numbers of case-specific sub-structures within the puzzle structure, indicating that specific types of sub-structure hinder recovery to such an extent that the scheme is infeasible for practical erasure correction. The consequences of complex cell inter-relationships and wider issues with puzzle-based erasure correction, beyond the structures investigated in the thesis are also discussed, concluding that while there are suggestions in the literature that Sudoku and other puzzle-based combinatorial structures may be useful for erasure correction, the work of this thesis suggests that this is not the case.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:749703 |
| Date | January 2013 |
| Creators | Phillips, Linzy |
| Contributors | Roach, Paul |
| Publisher | University of South Wales |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://pure.southwales.ac.uk/en/studentthesis/erasurecorrecting-codes-derived-from-sudoku--related-combinatorial-structures(b359130e-bfc2-4df0-a6f5-55879212010d).html |
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