Variable-rate linear network codes are investigated in this paper, which are referred to as linear network codes that can support a demanded range of transmission rates on a common netowrk. A new kind of linear network code, called as strict linear broadcast, is defined. Compared with general linear broadcast, it imposes more rigid constraints on the global encoding kernels, but does not require larger finite field size for construction. Then, an efficient scheme is proposed to construct variable-rate linear broadcasts based on the strict linear broadcast. Instead of construcing a fix-rate linear broadcast for each demanded transmission rate, this scheme implements variable-rate linear broadcasts with a single-rate strict linear broadcast. Every node in the network, including the source node, needs to store only one local encoding kernel. When transmission rate varies, the coding operations performed on every network node remain unchanged. Thus, small storage space and no kernel-swithching operations are required on any network code. Furthermore, by combining the strict linear broadcast with a special source-data packetization strategy, a hierarchical broadcast scheme is proposed. With this scheme, multi-rate service can be provided by a single-rate strict linear broadcast to heterogeneous receivers, even at variable transmission rate. Thus, the variable-rate linear broadcasts constructed in this paper are also applicable to the network with heterogeneous receivers.

Imaging the acoustical nonlinear parameter using second-harmonic and difference-frequency waves produced from finite-amplitude sound propagation is described. Experiments for reconstructing nonlinear-parameter tomograms were performed with gelatine phantoms. The tomograms were not of high quality because we used a small-aperture receiver to detect the substantially refracted waves. We show that by laterally translating the receiver to the position of maximum intensity for each measurement of a point on a projection, we can improve the quality of nonlinear parameter tomograms as well as that of conventional attenuation tomograms. We analyze the effects of attenuation and diffraction for a narrow transmitted beam in nonlinear parameter imaging. Using second-harmonic waves is advantageous for attenuation compensation when the attenuation in the object is proportional to the frequency, such as is the case for soft biological tissue. On the other hand, using difference-frequency waves has certain advantages where attenuation is not a problem. With these waves there is less phase variation.