Communication over Finite-Chain-Ring Matrix Channels

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over certain finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-ring matrix channel $Y = AX + BE$, where $X$ is the channel input, $Y$ is the channel output, $E$ is random error, and $A$ and $B$ are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under the assumption that $A$ is uniform over all full-rank matrices and $BE$ is uniform over all rank-$t$ matrices, extending the work of Silva, Kschischang and K\"{o}tter (2010), who handled the case of finite fields. This extension is based on several new results, which may be of independent interest, that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings.Comment: Submitted to IEEE Transactions on Information Theory, April 2013. Revised version submitted in Feb. 2014. Final version submitted in June 201

A global definition of quasinormal modes for Kerr-AdS Black Holes

The quasinormal frequencies of massive scalar fields on Kerr-AdS black holes are identified with poles of a certain meromorphic family of operators, once boundary conditions are specified at the conformal boundary. Consequently, the quasinormal frequencies form a discrete subset of the complex plane and the corresponding poles are of finite rank. This result holds for a broad class of elliptic boundary conditions, with no restrictions on the rotation speed of the black hole.Comment: 37 pages; minor changes. To appear in Ann. Inst. Fourie

Diophantine approximation and deformation

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree

Low-Rank Parity-Check Codes over Galois Rings

Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.Comment: 37 pages, 1 figure, extended version of arXiv:2001.0480