Linear codes are considered over the ring $mathbb{Z}_4+vmathbb{Z}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of Euclidean isod
ual codes, unimodular complex lattices, MDS codes and MGDS codes over $mathbb{Z}_4+vmathbb{Z}_4$ are studied. Cyclic codes and quadratic residue codes are also considered. Finally, some examples for illustrating the main work are given.
Based on the photometric data from the Spitzer/SAGE survey and with red giants as the extinction tracers, the mid-infrared (MIR) extinction laws in the Large Magellanic Cloud (LMC) are derived for the first time in the form of A_lambda/A_Ks, the exti
nction in the four IRAC bands (i.e., [3.6], [4.5], [5.8] and [8.0]um) relative to the 2MASS Ks band at 2.16um. We obtain the near-infrared (NIR) extinction coefficient to be E(J-H)/E(H-Ks)=1.29pm0.04 and E(J-Ks)/E(H-Ks)=1.94pm0.04. The wavelength dependence of the MIR extinction A_lambda/A_Ks in the LMC varies from one sightline to another. The overall mean MIR extinction is A_[3.6]/A_Ks=0.72pm0.03, A_[4.5]/A_Ks=0.94pm0.03, A_[5.8]/A_Ks=0.58pm0.04, and A_[8.0]/A_Ks=0.62pm0.05. Except for the extinction in the IRAC [4.5] band which may be contaminated by the 4.6um CO gas absorption of red giants (which are used to trace the LMC extinction), the extinction in the other three IRAC bands show a flat curve, close to the Milky Way Rv = 5.5 model extinction curve (where Rv is the optical total-to-selective extinction ratio). The possible systematic bias caused by the correlated uncertainties of Ks-lambda and J-Ks is explored in terms of Monte-Carlo simulations. It is found that this could lead to an overestimation of A_lambda/A_Ks in the MIR.
We have invented a novel technique to measure the radio image of a pulsar scattered by the interstellar plasma with 0.1 mas resolution. We extend the secondary spectrum analysis of parabolic arcs by Stinebring et al. (2001) to very long baseline inte
rferometry and, when the scattering is anisotropic, we are able to map the scattered brightness astrometrically with much higher resolution than the diffractive limit of the interferometer. We employ this technique to measure an extremely anisotropic scattered image of the pulsar B0834+06 at 327 MHz. We find that the scattering occurs in a compact region about 420 pc from the Earth. This image has two components, both essentially linear and nearly parallel. The primary feature, which is about 16 AU long and less than 0.5 AU in width, is highly inhomogeneous on spatial scales as small as 0.05 AU. The second feature is much fainter and is displaced from the axis of the primary feature by about 9 AU. We find that the velocity of the scattering plasma is 16+-10 km/s approximately parallel to the axis of the linear feature. The origin of the observed anisotropy is unclear and we discuss two very different models. It could be, as has been assumed in earlier work, that the turbulence on spatial scales of ~1000 km is homogeneous but anisotropic. However it may be that the turbulence on these scales is homogeneous and isotropic but the anisotropy is produced by highly elongated (filamentary) inhomogeneities of scale 0.05-16 AU.
Based on the data obtained from the Spitzer/GLIPMSE Legacy Program and the 2MASS project, we derive the extinction in the four IRAC bands, [3.6], [4.5], [5.8] and [8.0] micron, relative to the 2MASS Ks band (at 2.16 micron) for 131 GLIPMSE fields alo
ng the Galactic plane within |l|<65 deg, using red giants and red clump giants as tracers. As a whole, the mean extinction in the IRAC bands (normalized to the 2MASS Ks band), A_[3.6]/A_Ks=0.63, A_[4.5]/A_Ks=0.57, A_[5.8]/A_Ks=0.49, A_[8.0]/A_Ks=0.55, exhibits little variation with wavelength (i.e. the extinction is somewhat flat or gray). This is consistent with previous studies and agrees with that predicted from the standard interstellar grain model for R_V=5.5 by Weingartner & Draine (2001). As far as individual sightline is concerned, however, the wavelength dependence of the mid-infrared interstellar extinction A_{lambda}/A_Ks varies from one sightline to another, suggesting that there may not exist a universal IR extinction law. We, for the first time, demonstrate the existence of systematic variations of extinction with Galactic longitude which appears to correlate with the locations of spiral arms as well as with the variation of the far infrared luminosity of interstellar dust.
Observationally, both the 3.4micron aliphatic hydrocarbon C--H stretching absorption feature and the 9.7micron amorphous silicate Si--O stretching absorption feature show considerable variations from the local diffuse interstellar medium (ISM) to Gal
actic center (GC): both the ratio of the visual extinction (A_V) to the 9.7micron Si--O optical depth (tausil) and the ratio of A_V to the 3.4micron C--H optical depth (tauahc) of the solar neighborhood local diffuse ISM are about twice as much as that of the GC. In this work, we try to explain these variations in terms of a porous dust model consisting of a mixture of amorphous silicate, carbonaceous organic refractory dust (as well as water ice for the GC dust).
