The dielectric and other properties of a series of low-density polyimide block copolymer aerogels have been characterized. Two different anhydride-capped polyimide oligomers were synthesized: one from 2,2-bis(3,4-dicarboxyphenyl)hexafluoropropane dianhydride (6FDA) and 4,4'-oxidianiline (ODA) and the other from biphenyl-3,3',4,4'-tetracarboxylic dianhydride and ODA. The oligomers were combined with 1,3,5-triaminophenoxybenzene to form a block copolymer networked structure that gelled in under 1 h. The polyimide gels were supercritically dried to give aerogels with relative dielectric constants as low as 1.08. Increasing the amount of 6FDA blocks by up to 50% of the total dianhydride decreased the density of the aerogels, presumably by increasing the free volume and also by decreasing the amount of shrinkage seen upon processing, resulting in a concomitant decrease in the dielectric properties. In this study, we have also altered the density independent of fluorine substitution by changing the polymer concentration in the gelation reactions and showed that the change in dielectric due to density is the same with and without fluorine substitution. The aerogels with the lowest dielectric properties and lowest densities still had compressive moduli of 4-8 MPa (40 times higher than silica aerogels at the same density), making them suitable as low dielectric substrates for lightweight antennas for aeronautic and space applications.
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and leftregular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.
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