Abstract
In this paper we present generalized edge-pairings for the family of hyperbolic tessellations \(\{10\lambda ,2\lambda \}\), with the purpose to obtain the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations \(\{4\lambda ,4\lambda \}\) implying that the associated codes achieve the least error probability, or equivalently, that these codes are optimum codes.
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Notes
\(SU(2,\mathbb {C})\) is the special unitary group.
The stabilizer of an element \(z\in \mathbb {D}^{2}\) is the subgroup \(G_{z}=\{T\in \varGamma _{p}:T(z)=z\}\).
If the condition (21) is not satisfied, then one can not guarantee that the group \(\varGamma _{p}\) is discrete.
This implies that the internal angles at the corresponding vertices of the polygon \(P_{p}\) is \(\frac{2\pi }{q}=\frac{\pi }{\lambda }\).
For \(\lambda =4\), we order \(C_{2}\) in the form \(C_{2}=\{v_{5},\) \(v_{10},\) \(v_{15},\) \(v_{20},\) \(v_{25},\) \(v_{30},\) \(v_{35},\) \(v_{40}\}\).
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Acknowledgments
This work has been supported by: PROPESQ/UEPB, under grant 02/2010, FAPESP under grant 04/15328-2 and under grant 2007/56052-8, CNPq under grant 505258/2008-0 and under grant 303059/2010-9, under grant UFV.
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Communicated by Yoshiharu Kohayakawa.
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Vieira, V.L., Faria, M.B. & Palazzo, R. Generalized edge-pairings for the family of hyperbolic tessellations \(\{10\lambda ,2\lambda \}\) . Comp. Appl. Math. 35, 29–43 (2016). https://doi.org/10.1007/s40314-014-0178-z
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DOI: https://doi.org/10.1007/s40314-014-0178-z