On a class of optimal rateless codes

Abstract

Abstract-In this paper we analyze a class of systematic fountain/rateless codes constructed using Bernoulli(1/2) random variables. Using simple bounds we then show that this class of codes stochastically minimizes the number of coded packets receptions needed to successfully decode all the information packets. This optimality holds over a large class of random codes that includes Bernoulli(q) random codes with q ≤ 1/2 and LT codes. We then conclude by demonstrating asymptotic optimality for intermediate decoding of the same codes.

Message Forwarding in Sparsely Connected Wireless Networks Using Rateless Codes

“…The particular rateless code used in this paper is based on [3]. It is possible to analytically obtain costs and other useful quantities explicitly for this.…”

“…Rateless codes use such a channel efficiently [1][2][3][4] with minimal feedback. Linear combinations of the packets of a multi-packet message are sent until enough linearly independent combinations are received to decode each packet and hence the overall message.…”

Low-delay dynamic routing using fountain codes

“…That is, each element in vector g i takes value 1 with probability 0.5 (and so also takes value 0 with probability 0.5, hence 0 and 1 are equiprobable). It is shown in [9] that this class of fountain codes has lower overhead δ (and so decoding delay) than any sparse code. Use of a systematic code further reduces decoding delay when the level of packet reordering is low.…”

PHY modulation/rate control for fountain codes in 802.11a/g WLANs

“…In the following we assume that the acknowledgement packet has a 8 byte payload, and is transmitted at the basic PHY rate of 6 Mbps. We use a fountain code block size N of 50 packets and assume an overhead δ of 2 packets for both systematic and non-systematic fountain codes (this is a lower bound on the decoding overhead, see [18])-unless otherwise stated we also use these values of N and δ in the remainder of the paper. At low PERs, uncoded and systematic fountain coded traffic have similar goodputs and this is higher than the goodput with a non-systematic fountain code owing to the decoding overhead δ.…”

“…Source packets are selected in terms of a Bernoulli (1/2) random vector and then summed, modulo 2, to construct a coded packet. The systematic fountain code we consider in the numerical results is proposed in [18], which uses uncoded source packets for the first N transmissions and the subsequent transmissions are coded packets constructed using an equiprobable random linear fountain code. It is shown in [18] that this class of fountain codes stochastically minimizes the number of received packets necessary for recovery of coded packets over a large class of fountain codes including Raptor and LT codes, and has low decoding complexity for small block sizes.…”