Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space $V$ and the collection by the receiver of a basis for a vector space $U$. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space $V \cap U$ is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification

Dealing with Incomplete Household Panel Data in Inequality Research

Population surveys around the world face the problem of declining cooperation and participation rates of respondents. Not only can item nonresponse and unit nonresponse impair important outcome measures for inequality research such as total household disposable income; there is also a further case of missingness confronting household panel surveys that potentially biases results. The approach commonly used in such surveys of interviewing all adult household members and aggregating their individual incomes to yield a final outcome measure for welfare analyses often suffers from partial unit non-response (PUNR), i.e., the non-response of at least one unit, or member, of an otherwise participating household. In these cases, the aggregate income of all household members lacks at least one individual's income. These processes are typically not random and require appropriate correction. Using data from more than twenty waves of the German Socio-Economic Panel (SOEP) we evaluate four different strategies to deal with this phenomenon: (a) Ignorance, i.e., assuming the missing individual's income to be zero. (b) Adjustment of the equivalence scale to account for differences in household size and composition. (c) Elimination of all households observed to suffer PUNR, and re-weighting of households observed to be at risk of but not affected by PUNR. (d) Longitudinal imputation of the missing income components. The aim of this paper is to show how the choice of technique affects substantive results in the inequality research. We find indications of substantial bias on income inequality and poverty as well as on income mobility. These findings are obviously even more important in cross-national comparative analyses if the data providers in the individual countries deal differently with PUNR in the underlying data.Household Panel Surveys, Partial Unit Non-Response, Inequality, Mobility, Imputation, SOEP