As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper,...As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.展开更多
For accurate and stable haptic rendering, collision detection for interactive haptic applications has to be done by filling in or covering target objects as tightly as possible with bounding volumes （spheres, axis-al...For accurate and stable haptic rendering, collision detection for interactive haptic applications has to be done by filling in or covering target objects as tightly as possible with bounding volumes （spheres, axis-aligned bounding boxes, oriented bounding boxes, or polytopes）. In this paper, we propose a method for creating bounding spheres with respect to the contact levels of details （CLOD）, which can fit objects while maintaining the balance between high speed and precision of collision detection. Our method is composed mainly of two parts： bounding sphere formation and two-level collision detection. To specify further, bounding sphere formation can be divided into two steps： creating spheres and clustering spheres. Two-level collision detection has two stages as well： fast detection of spheres and precise detection in spheres. First, bounding spheres are created for initial fast probing to detect collisions of spheres. Once a collision is probed, a more precise detection is executed by examining the distance between a haptie pointer and each mesh inside the colliding boundaries. To achieve this refmed level of detection, a special data structure of a bounding volume needs to be defined to include all mesh information in the sphere. After performing a number of experiments to examine the usefulness and performance of our method, we have concluded that our algorithm is fast and precise enough for haptic simulations. The high speed detection is achieved through the clustering of spheres, while detection precision is realized by voxel-based direct collision detection. Our method retains its originality through the CLOD by distance-based clustering.展开更多
Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)...Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.展开更多
文摘As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.
基金supported by Incheon National University Research,Korea(No.20120238)
文摘For accurate and stable haptic rendering, collision detection for interactive haptic applications has to be done by filling in or covering target objects as tightly as possible with bounding volumes （spheres, axis-aligned bounding boxes, oriented bounding boxes, or polytopes）. In this paper, we propose a method for creating bounding spheres with respect to the contact levels of details （CLOD）, which can fit objects while maintaining the balance between high speed and precision of collision detection. Our method is composed mainly of two parts： bounding sphere formation and two-level collision detection. To specify further, bounding sphere formation can be divided into two steps： creating spheres and clustering spheres. Two-level collision detection has two stages as well： fast detection of spheres and precise detection in spheres. First, bounding spheres are created for initial fast probing to detect collisions of spheres. Once a collision is probed, a more precise detection is executed by examining the distance between a haptie pointer and each mesh inside the colliding boundaries. To achieve this refmed level of detection, a special data structure of a bounding volume needs to be defined to include all mesh information in the sphere. After performing a number of experiments to examine the usefulness and performance of our method, we have concluded that our algorithm is fast and precise enough for haptic simulations. The high speed detection is achieved through the clustering of spheres, while detection precision is realized by voxel-based direct collision detection. Our method retains its originality through the CLOD by distance-based clustering.
基金This work was supported in part by the National Natural Science Foundation of China(NSFC)under Grants 11971156 and 12001175.
文摘Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.
