Abstract
Let \({\mathcal{L}}\) and \({\mathcal{L}_0}\) be the binary codes generated by the column \({\mathbb{F}_2}\)-null spaces of the incidence matrices of external points versus passant lines and internal points versus secant lines with respect to a conic in PG(2, q), respectively. We confirm the conjectures on the dimensions of \({\mathcal{L}}\) and \({\mathcal{L}_0}\) using methods from both finite geometry and modular representation theory.
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Communicated by Q. Xiang.
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Wu, J. Proofs of two conjectures on the dimensions of binary codes. Des. Codes Cryptogr. 70, 273–304 (2014). https://doi.org/10.1007/s10623-012-9682-6
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DOI: https://doi.org/10.1007/s10623-012-9682-6
Keywords
- Block idempotent
- Brauer’s theory
- Character
- Conic
- General linear group
- Incidence matrix
- Low-density parity-checkcode
- Module
- 2-Rank