Generalized weights: an anticode approach

In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then we characterize both generalized Hamming and rank weights of a code in terms of the intersection of the code with optimal anticodes in the respective metrics. Inspired by this description, we propose a new algebraic invariant, which we call "Delsarte generalized weights", for Delsarte rank-metric codes based on optimal anticodes of matrices. We show that our invariant refines the generalized rank weights for Gabidulin codes proposed by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of Delsarte generalized weights. In particular, we characterize Delsarte optimal codes and anticodes in terms of their generalized weights. We also present a duality theory for the new algebraic invariant, proving that the Delsarte generalized weights of a code completely determine the Delsarte generalized weights of the dual code. Our results extend the theory of generalized rank weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes of a theorem of Wei, proving that their generalized rank weights characterize the worst-case security drops of a Gabidulin rank-metric code

Generalized selfish bin packing

Standard bin packing is the problem of partitioning a set of items with positive sizes no larger than 1 into a minimum number of subsets (called bins) each having a total size of at most 1. In bin packing games, an item has a positive weight, and given a valid packing or partition of the items, each item has a cost or a payoff associated with it. We study a class of bin packing games where the payoff of an item is the ratio between its weight and the total weight of items packed with it, that is, the cost sharing is based linearly on the weights of items. We study several types of pure Nash equilibria: standard Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and weakly Pareto optimal equilibria. We show that any game of this class admits all these types of equilibria. We study the (asymptotic) prices of anarchy and stability (PoA and PoS) of the problem with respect to these four types of equilibria, for the two cases of general weights and of unit weights. We show that while the case of general weights is strongly related to the well-known First Fit algorithm, and all the four PoA values are equal to 1.7, this is not true for unit weights. In particular, we show that all of them are strictly below 1.7, the strong PoA is equal to approximately 1.691 (another well-known number in bin packing) while the strictly Pareto optimal PoA is much lower. We show that all the PoS values are equal to 1, except for those of strong equilibria, which is equal to 1.7 for general weights, and to approximately 1.611824 for unit weights. This last value is not known to be the (asymptotic) approximation ratio of any well-known algorithm for bin packing. Finally, we study convergence to equilibria

Optimal Prefix Codes with Fewer Distinct Codeword Lengths are Faster to Construct

A new method for constructing minimum-redundancy binary prefix codes is described. Our method does not explicitly build a Huffman tree; instead it uses a property of optimal prefix codes to compute the codeword lengths corresponding to the input weights. Let $n$ be the number of weights and $k$ be the number of distinct codeword lengths as produced by the algorithm for the optimum codes. The running time of our algorithm is $O(k \cdot n)$. Following our previous work in \cite{be}, no algorithm can possibly construct optimal prefix codes in $o(k \cdot n)$ time. When the given weights are presorted our algorithm performs $O(9^k \cdot \log^{2k}{n})$ comparisons.Comment: 23 pages, a preliminary version appeared in STACS 200

Overcoming device unreliability with continuous learning in a population coding based computing system

The brain, which uses redundancy and continuous learning to overcome the unreliability of its components, provides a promising path to building computing systems that are robust to the unreliability of their constituent nanodevices. In this work, we illustrate this path by a computing system based on population coding with magnetic tunnel junctions that implement both neurons and synaptic weights. We show that equipping such a system with continuous learning enables it to recover from the loss of neurons and makes it possible to use unreliable synaptic weights (i.e. low energy barrier magnetic memories). There is a tradeoff between power consumption and precision because low energy barrier memories consume less energy than high barrier ones. For a given precision, there is an optimal number of neurons and an optimal energy barrier for the weights that leads to minimum power consumption

Optimizing the new formulation of the United Nations' human development index: An empirical view from data envelopment analysis.

In this paper, we propose a new way to simulate an optimal Human Development Index [HDI]. Indeed, the formulation of the original HDI established by the United Nations Development Programme [UNDP] relies on a major methodological shortcoming, namely the contestable assumption that all component indices have the same weights. So, we implement a new approach to determine the optimal weights of each sub-indicator in the light of Data Envelopment Analysis [DEA]. Accordingly, we follow the multiplicative optimization approach introduced by Zhou et al. (2010), to assess robustly the relative performance of a set of 169 economies around the world in terms of human development. Finally, the new world ranking is close to and highly correlated with the standard HDI one, giving then some support to the equal weighting method adopted by the UNDP.Human development index, data envelopment analysis, multiplicative optimization approach, optimal weights