On a Relationship between Pascal Matrix and Vandermonde Matrix

EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.

A NOVEL FRAMEWORK OF ONLINE NETWORK CODING FOR MULTICAST SWITCHES WITH CONSTRAINED BUFFERS

Network coding is able to address output conflicts when fanout splitting is allowed for multicast switching.Hence,it successfully achieves a larger rate region than non-coding approaches in crossbar switches.However,network coding requires large coding buffers and a high computational cost on encoding and decoding.In this paper,we propose a novel Online Network Coding framework called Online NC for multicast switches,which is adaptive to constrained buffers.Moreover,it enjoys a much lower decoding complexity by a Vandermonde matrix based approach,as compared to conven-tional randomized network coding Our approach realizes online coding with one coding algo-rithm that synchronizes buffering and coding.Therefore,we significantly reduce requirements on buffer space,while also sustaining high throughputs.We confirm the superior advantages of our contributions using empirical studies.

The Exact Value of det V - n(x 1, ...,x n) and Its Applications

Suppose that x is a complex number and i is a non negative integer. Define N - i(x)=|x| i if i is even and N - i(x)=x|x| i-1 if i is odd. Let V - n(x 1, ...,x n) denote the n× n matrix whose (i,j) th entry is N - i-1 (x j) . This paper presents a computation formula for det V - n(x 1, ...,x n) , which can be considered as a generalized that of Vandermonde determinant, and some its important theoretical applications.

Efficient wireless multicast retransmission using network coding

In wireless multicast, network coding has recently attracted attentions as a substantial improvement to packet retransmission schemes. However, the problem of finding the optimal network code which minimizes the retransmissions is hard to solve or approximate. This paper presents two schemes to reduce the number of retransmissions for reliable multicast efficiently. One is retransmission using network coding based on improved Vandermonde matrix （VRNC）, the other is retransmission using network coding based on adaptive improved Vandermonde matrix （AVRNC）. Using VRNC scheme the sender selects the packets all receivers have lost and encodes them with improved Vandermonde matrix; when receivers receive enough encoded retransmission packets, all the lost packets can be recovered. With AVRNC scheme, the sender can obtain the recovery information from all the receivers after sending out per retransmission packet, and then the improved Vandermonde matrix can be updated, thus reducing the complexity of encoding and decoding. Our proposed schemes can achieve the theoretical lower bound assuming retransmission packets lossless, and approach the theoretical lower bound considering retransmission packets loss. Simulation results show that the proposed algorithms can efficiently reduce the number ofretransmissions, thus improving transmission efficiency.