Rateless Codes with Optimum Intermediate Performance Ali Talari and Nazanin Rahnavard School of Electrical and Computer Engineering Oklahoma State University Stillwater, OK 74078 Emails: {ali.talari, nazanin.rahnavard}@okstate.edu Abstract—In this paper, we design several degree distributions for rateless codes with optimum intermediate packet recovery rates. In rateless coding, the employed degree distribution signif- icantly affects the packet recovery rate. Each degree distribution is designed based on the number of message packets, k, and desired coding overhead, γ, which is the ratio of the number of received packets, n, to k, i.e., γ = n k . Previously designed degree distributions are tuned for full recovery of the entire source packets for γ’s slightly larger than 1, and as a consequence, they show very small packet recovery rates for γ< 1. Hence, finding degree distributions with maximal packet recovery rates in intermediate range, 0 <γ< 1, is of interest. We define packet recovery rates at three values of γ as our conflicting objective functions and employ NSGA-II multi- objective genetic algorithms optimization method to find several degree distributions with optimum packet recovery rates. We propose degree distributions for both cases of finite and infinite (asymptotic) k. I. I NTRODUCTION Intermediate recovery rate is important in applications where partial recovery of the source packets from received encoded packets is still beneficial. For instance, in video or voice transmission, the receiver can benefit from incomplete recovered data by playing a lower quality of the media. This motivates the design of forward error correction (FEC) codes with high intermediate performance. Here, the term performance refers to the the ability of the employed coding scheme to recover the maximum number of source packets at the receiver. Rateless codes are modern flexible FEC codes with low cod- ing/decoding complexity. Since rateless codes do not impose a fixed coding rate, they can be employed on channels with varying or unknown loss rates. Previously designed rateless codes [1–3] offer efficient FEC codes for recovery of the entire message with low error probability while they show a weak performance in intermediate range. Therefore, it is of interest to design rateless codes with optimum intermediate performance. The parameter that determines the packet recovery rate of rateless codes is the employed coding degree distribution. In rateless encoding, first a packet degree, d, is chosen from a degree distribution, {Ω 1 , Ω 2 ,..., Ω n }, where Ω i is the proba- bility that d = i and ∑ n i=1 Ω i =1. This degree distribution is also shown by its generator polynomial Ω(x)= ∑ n i=1 Ω i x i . Next d source packets are chosen uniformly at random from the source packets, and are XORed together to generate an encoded packet. This procedure is repeated until enough number of packets is collected at the receiver. In this paper, we denote the number of source packets by k, number of received coded packets by n, and received overhead by γ , where γ = n k . Furthermore, we denote the ratio of number of recovered packets at the receiver to k by z . The decoding of rateless codes is performed in an iterative fashion. First, the decoder finds received coded packets of degree-one and recovers one source packet from each one of these coded packets. Next, by removing the recovered source packets from higher degree encoded packets more degree-one coded packets emerge. This iterative decoding continues until no more coded packets can be reduced to degree one. The degree distribution of rateless codes are usually finely tuned to get the optimum results at fixed γ ’s slightly larger than one, while here we prefer to have codes with superior performance in all intermediate range, 0 <γ< 1. Note that, when the packet recovery rate at a certain γ is increased by modifying the degree distribution, the recovery rate at other γ ’s degrades. In other words, we have to deal with multiple dependant recovery rates. Consequently, if we consider recov- ery rates as conflicting objective functions, we have a multi- objective optimization problem. Later we show that based on previous studies [4] the overhead range of 0 <γ< 1 is divided into three regions. Therefore, we choose one fixed γ from each region, i.e. γ ∈{0.5, 0.75, 1}, and employ a multi-objective optimization method to find the degree distribution with maximal packet recovery rates at these three γ ’s. Each γ represents one region of intermediate range and the resulting degree distributions would perform optimally throughout intermediate range. Since we have three objective functions, the optimum answers will be a member of a 3D pareto front [5] in our objective space. In this 3D space, each dimension represents an objective function. Our decision space includes several continuous coefficients, Ω i , of each degree. Since our decision space is huge, we propose to employ the state of the art multi-objective genetic algorithm NSGA-II [5]. When the pareto front members are found, we define a total cost which is the weighted summation of difference of