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

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

Relationships between the perceived quality of life and the personality styles measured with the The Millon Index of Personality Styles Revised (MIPS-R)

This exploratory study aims to determine whether the personality styles measured with the Portuguese adaptation of Millon Index of Personality Styles Revised, MIPS-R affect the perceived quality of life. The MIPS-R is a theory-based inventory that measures 24 personality styles in normally functioning adults. Life satisfaction was measured with the Portuguese version of the Quality of Life Inventory, QOLI (Fagulha, Duarte & Miranda, 2000). It refers to a person’s subjective evaluation of the degree to which his/her most important needs, goals and wishes have been fulfilled. This study was carried out with a sample of 43 college students, 36 females (age mean = 19,7; SD = 3,1) and 7 males (age mean = 27,4; SD = 11,4). Based on the participants’ overall life satisfaction score three groups were defined: (1) Low/Very Low quality of life, (2) Average quality of life, (3) High quality of life. Discriminant Factor Analysis (DFA) and the Kruskal-Wallis Test were used to identify the styles that most differentiate these groups and to compare each style in the groups. The Other-Nurturing style is the one that best differentiates the groups. DFA results will be further exploited. Considering the Kruskal-Wallis Test, differences are observed in the Pleasure-Enhancing (p=.006), the Actively Modifying (p=.002), the Gregarious/Outgoing (p=.012), the Passively Accommodating (p=.027), the Asocial/Withdrawing (p=.036), the Unconventional/Dissenting (p=.041) and in the Dissatisfied/Complaining (p=.019) styles. Multiple comparisons were used to compare these styles in the groups. The authors believe that the discussion of these results will provide a better understanding of the MIPS-R.Instituto de Psicologia das Relações Humana

Security for Wiretap Networks via Rank-Metric Codes

The problem of securing a network coding communication system against a wiretapper adversary is considered. The network implements linear network coding to deliver $n$ packets from source to each receiver, and the wiretapper can eavesdrop on $\mu$ arbitrarily chosen links. A coding scheme is proposed that can achieve the maximum possible rate of $k=n-\mu$ packets that are information-theoretically secure from the adversary. A distinctive feature of our scheme is that it is universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying network code. In fact, even a randomized network code can be used. Our approach is based on Rouayheb-Soljanin's formulation of a wiretap network as a generalization of the Ozarow-Wyner wiretap channel of type II. Essentially, the linear MDS code in Ozarow-Wyner's coset coding scheme is replaced by a maximum-rank-distance code over an extension of the field in which linear network coding operations are performed.Comment: 5 pages, to be published at the 2008 IEEE International Symposium on Information Theor