Computing the Ball Size of Frequency Permutations under Chebyshev Distance

Let $S_n^\lambda$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^{\lambda},...,\overbrace{m,...,m}^\lambda\}$ where $n=m\lambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^\lambda$ in which every two elements have distance at least $d$. FPAs have many applications related to error correcting codes. In coding theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived from the size of balls of certain radii. We propose two efficient algorithms that compute the ball size of frequency permutations under Chebyshev distance. Both methods extend previous known results. The first one runs in $O({2d\lambda \choose d\lambda}^{2.376}\log n)$ time and $O({2d\lambda \choose d\lambda}^{2})$ space. The second one runs in $O({2d\lambda \choose d\lambda}{d\lambda+\lambda\choose \lambda}\frac{n}{\lambda})$ time and $O({2d\lambda \choose d\lambda})$ space. For small constants $\lambda$ and $d$, both are efficient in time and use constant storage space.Comment: Submitted to ISIT 201

Fractal model and Lattice Boltzmann Method for Characterization of Non-Darcy Flow in Rough Fractures.

The irregular morphology of single rock fracture significantly influences subsurface fluid flow and gives rise to a complex and unsteady flow state that typically cannot be appropriately described using simple laws. Yet the fluid flow in rough fractures of underground rock is poorly understood. Here we present a numerical method and experimental measurements to probe the effect of fracture roughness on the properties of fluid flow in fractured rock. We develop a series of fracture models with various degrees of roughness characterized by fractal dimensions that are based on the Weierstrass-Mandelbrot fractal function. The Lattice Boltzmann Method (LBM), a discrete numerical algorithm, is employed for characterizing the complex unsteady non-Darcy flow through the single rough fractures and validated by experimental observations under the same conditions. Comparison indicates that the LBM effectively characterizes the unsteady non-Darcy flow in single rough fractures. Our LBM model predicts experimental measurements of unsteady fluid flow through single rough fractures with great satisfactory, but significant deviation is obtained from the conventional cubic law, showing the superiority of LBM models of single rough fractures