On the Construction of High Dimensional Simple Games

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. However, its naive encoding needs $2^n$ bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using $n$ weights and one threshold. For heterogeneous agents, one can represent $\chi$ as an intersection of $k$ threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring $k\ge 2^{\frac{n}{2}-1}$ and provided a construction guaranteeing $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-o(n)}$. The magnitude of the worst-case situation was thought to be determined by Elkind et al.~in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number $k$ for a subclass of voting systems. As an application, we give a construction for $k\ge 2^{n-o(n)}$, i.e., there is no gain from a representation complexity point of view.Comment: 13 pages, 1 tabl

Asymptotics for the genus and the number of rational places in towers of function fields over a finite field

We discuss the asymptotic behaviour of the genus and the number of rational places in towers of function fields over a finite field

Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires list size $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires list size $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values

On Weak Odd Domination and Graph-based Quantum Secret Sharing

A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. We point out the connections of weak odd domination with odd domination, [sigma,rho]-domination, and perfect codes. We introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graph-based quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a \kappa_Q smaller than 0.811n.Comment: Subsumes arXiv:1109.6181: Optimal accessing and non-accessing structures for graph protocol

Synchronization with permutation codes and Reed-Solomon codes

D.Ing. (Electrical And Electronic Engineering)We address the issue of synchronization, using sync-words (or markers), for encoded data. We focus on data that is encoded using permutation codes or Reed-Solomon codes. For each type of code (permutation code and Reed-Solomon code) we give a synchronization procedure or algorithm such that synchronization is improved compared to when the procedure is not employed. The gure of merit for judging the performance is probability of synchronization (acquisition). The word acquisition is used to indicate that a sync-word is acquired or found in the right place in a frame. A new synchronization procedure for permutation codes is presented. This procedure is about nding sync-words that can be used speci cally with permutation codes, such that acceptable synchronization performance is possible even under channels with frequency selective fading/jamming, such as the power line communication channel. Our new procedure is tested with permutation codes known as distance-preserving mappings (DPMs). DPMs were chosen because they have de ned encoding and decoding procedures. Another new procedure for avoiding symbols in Reed-Solomon codes is presented. We call the procedure symbol avoidance. The symbol avoidance procedure is then used to improve the synchronization performance of Reed-Solomon codes, where known binary sync-words are used for synchronization. We give performance comparison results, in terms of probability of synchronization, where we compare Reed-Solomon with and without symbol avoidance applied