A Rank-Metric Approach to Error Control in Random Network Coding

The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if $\mu$ erasures and $\delta$ deviations occur, then errors of rank $t$ can always be corrected provided that $2t \leq d - 1 + \mu + \delta$, where $d$ is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where $n$ packets of length $M$ over $F_q$ are transmitted, the complexity of the decoding algorithm is given by $O(dM)$ operations in an extension field $F_{q^n}$.Comment: Minor corrections; 42 pages, to be published at the IEEE Transactions on Information Theor

Algebraic Approach to Physical-Layer Network Coding

The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, non-asymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes---called lattice network coding (LNC) schemes for short---by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is re-interpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which non-coherent network coding can be achieved. Next, performance/complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum inter-coset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Construction A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.Comment: Submitted to IEEE Transactions on Information Theory, July 21, 2011. Revised version submitted Sept. 17, 2012. Final version submitted July 3, 201

Thermodynamics of small superconductors with fixed particle number

The Variation After Projection approach is applied for the first time to the pairing hamiltonian to describe the thermodynamics of small systems with fixed particle number. The minimization of the free energy is made by a direct diagonalization of the entropy. The Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.Comment: 4 pages, 3 figure

The Mass Growth and Stellar Ages of Galaxies: Observations versus Simulations

Using observed stellar mass functions out to $z=5$, we measure the main progenitor stellar mass growth of descendant galaxies with masses of $\log{M_{*}/M_{\odot}}=11.5,11.0,10.5,10.0$ at $z\sim0.1$ using an evolving cumulative number density selection. From these mass growth histories, we are able to measure the time at which half the total stellar mass of the descendant galaxy was assembled, $t_{a}$, which, in order of decreasing mass corresponds to redshifts of $z_{a}=1.28, 0.92, 0.60$ and $0.51$. We compare this to the median light-weighted stellar age $t_{*}$ ($z_{*} = 2.08, 1.49, 0.82$ and $0.37$) of a sample of low redshift SDSS galaxies (from the literature) and find the timescales are consistent with more massive galaxies forming a higher fraction of their stars ex-situ compared to lower mass descendants. We find that both $t_{*}$ and $t_{a}$ strongly correlate with mass which is in contrast to what is found in the EAGLE hydrodynamical simulation which shows a flat relationship between $t_{a}$ and $M_{*}$. However, the semi-analytic model of \citet{henriques2015} is consistent with the observations in both $t_{a}$ and $t_{*}$ with $M_{*}$, showing the most recent semi-analytic models are better able to decouple the evolution of the baryons from the dark matter in lower-mass galaxies.Comment: 6 pages, 3 figures, accepted for publication in ApJ