Weights which respect support and NNdecoding In this work we explore a family of metrics over finite fields which respect the support of vectors. We show how these metrics can be obtained from the edgeweighted Hamming cube and, based on this representation we give a description of a group of linear isometries (with respect to the metric). Next we introduce the concept of conditional sum of metrics and determine what conditions determine a metric respecting support, out of two such given metrics. Finally we introduce the labeledposet block metrics, a new family of metrics which respects support of vector, filling a gap existing in the known such metrics. For this family we give a full description of the group of linear isometries and determine necessary and sufficient conditions for the existence of a MacWilliams identity.

Revisiting Graph Width Measures for CNFEncodings We consider bounded width CNFformulas where the width is measured by popular graph width measures on graphs associated to CNFformulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNFformulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.