Dual code

We develop a Tanner graph (TG) construction for an Abelian group block code with arbitrary alphabets at different coordinates, an important application of which is the representation of the label code of a lattice. The construction is based on the modular linear constraints imposed on the code symbols by a set of generators for the dual code. As a necessary step toward the construction of a TG for , we devise an efficient algorithm for finding a generating set for. In the process, we develop a construction for lattices based on an arbitrary Abelian group block code, called generalized Construction A (GCA), and explore relationships among a group code, its GCA lattice, and their duals. We also study the problem of finding low-complexity TGs for Abelian group block codes and lattices, and derive tight lower bounds on the label-code complexity of lattices. It is shown that for many important lattices, the minimal label codes which achieve the lower bounds cannot be supported by cycle-free Tanner graphs.

An appropriate combination of the thin-layer Navier-Stokes (TLNS) and parabolized Navier-Stokes (PNS) solvers is used to accurately and efficiently compute hypersonic flowfields of equilibrium air around blunt-body configurations. The TLNS equations are solved in the nose region to provide the initial data plane needed for the solution of the PNS equations. Then the PNS equations are employed to efficiently compute the flowfield for the afterbody region by using a space marching procedure. Both the TLNS and PNS equations are numerically solved by using the efficient implicit non-iterative finite-difference algorithm of Beam and Warming. A shock fitting technique is used in both the TLNS and PNS codes to obtain accurate solution in the vicinity of the shock. In the regions of the flowfield with upstream influences, the globally iterated PNS (IPNS) equations, which are computationally more efficient than the TLNS equations, can be used to improve the accuracy of the results. To validate the results of the developed TLNS code and study real gas effects on the flowfield, hypersonic laminar flow over a sphere at Mach number of 11.26 is computed. The computations are performed for hypersonic laminar flow over 5 • long slender blunt cone at Mach number of 19.25 to demonstrate the accuracy and efficiency of using the present TLNS-PNS methodology. To show the capability and robustness of the present solution procedure, the comparisons are also made with the wall heat-transfer data on a 12.84 • /7 • bicone at Mach number of 6.89 and freestream temperature of 1604 K. The results of these computations are found to be in good agreement with available numerical and experimental data. The results of the perfect gas for these cases are also presented and the effects of real gas on the flowfield characteristics are studied in both the TLNS and PNS solutions.

A standard code for information interchange is required in order to provide a uniform means of communication among data processing equipments. Such standard codes have been long in existence for Latin languages. Unfortunately, establishing similar standard codes for non-Latin languages has been slow and difficult. Two such languages are Arabic and Farsi, the latter being the language of countries such as Iran, Afghanistan, and as Urdu language in Pakistan and India. The need for a standard code for computer data representation is well understood. Such standards for Latin speaking countries have been in existence for decades and have facilitated information interchange and the advancement of computer peripheral technology. In this paper we present the proposed Iranian Standard Code for Information Interchange (ISCII) and the proposed Iranian Standard Keyboard Layout. The Iranian standards are specified to accomodate the difficult goals of maintaining the 7-bit ASCII subset for bilingual applications and a well defined collating sequence while allowing for the variety of shapes that letters of the Farsi alphabet assume depending on their position in a written word, a problem that is shared by another script language-Arabic. The proposed ISCII, which is compatible with ISO-IRV, is shown to satisfy the two criteria of sufficiency and efficiency for computer data representation and display purposes. Furthermore, it is independent of the strengths or limitations of any particular computer output display technology.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.