IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 9, SEPTEMBER 2012 2377 Transactions Letters On Raptor Code Design for Inactivation Decoding Kaveh Mahdaviani, Student Member, IEEE, Masoud Ardakani, Senior Member, IEEE, and Chintha Tellambura, Fellow, IEEE Abstract—Based on a new vision of the inactivation decoding process, we set a new degree distribution design criterion for the LT part of Raptor codes. Under an infinite block length assumption, a family of degree distributions that satisfy the new design criterion is analytically derived. The finite length performance of this family is investigated by using computer simulations and is shown to outperform the conventional design. Index Terms—Finite length Raptor codes, inactivation decod- ing, Soliton distribution. I. I NTRODUCTION L UBY transform (LT) codes were originally introduced as the first practical fountain codes in [1]. As such, LT codes are designed to transmit a theoretically endless stream of symbols until the receiver has enough symbols to decode all the information bits. Raptor codes [2], an extension of LT codes, employ an outer code to enable the receiver to recover the whole information stream from any sufficiently large subset of recovered intermediate symbols. This idea significantly improves the performance of LT codes, as the recovery of the last few percentages of the information bits, which could be very slow, is now done by using the outer code. Raptor codes are able to asymptotically achieve the chan- nel capacity on any binary erasure channel (BEC) with- out any channel state information at the transmitter or the receiver. This universal capacity-achieving property enables optimal performance even in time-varying channels. Ac- cordingly, these codes are the natural choice for broadcast- ing/multicasting to a group of receivers with different and even unknown channel qualities. As a result Raptor codes have been adopted by the 3rd Generation Group Partnership Project (3GPP) to be used in multimedia broadcast/multicast services (MBMS) for forward error correction [3] and digital video broadcast-handheld (DVB-H) [4]. The desirable properties of Raptor codes have motivated many researchers to study their performance and design for other channels [5], [6]. Decoder design for Raptor codes has also been an active research area [7]–[9]. In this work, we study the design of Raptor codes for the BEC when inactivation decoding (ID) [9] is used. An ID Paper approved by T. M. Duman, the Editor for Coding Theory and Applications of the IEEE Communications Society. Manuscript received July 5, 2011; revised January 17 and April 18, 2012. The authors are with the Dept. of Elec. and Comp. Eng., University of Alberta (e-mail: {kmahdavi, ardakani, chintha}@ece.ualberta.ca). Digital Object Identifier 10.1109/TCOMM.2012.072612.110143 decoder is essentially a maximum likelihood decoder with controlled complexity, which can accomplish the decoding with a smaller number of received symbols than any other decoder requires. Hence, ID is incorporated in 3GPP as a practical decoder [3]. Despite the rich literature on code design for the conventional edge deletion decoding (e.g., [2], [10], [11]), code design for ID has not yet received much attention. In the remainder of this article, we first briefly review the encoding and decoding of Raptor codes, focusing on ID. In Section III, we introduce our code design, by proposing a new design criterion, and then we use this criterion for an analytical design. The numerical comparisons between the code used by the 3GPP and our proposed code are presented in Section IV. II. ENCODING AND DECODING OF RAPTOR CODES Encoding of the Raptor codes is done in two separate steps. In the first step, k information bits are coded to n = k/R intermediate bits by using an outer code of rate R. In the second step, the LT encoder first uses a probability distribution to choose an integer m ∈{1, ··· ,D}, D ≤ n, and then uniformly at random chooses m intermediate bits whose XOR forms an output symbol for transmission. The probability distribution of m is characterized by a generating polynomial Ω(x)= ∑ D i=1 Ω i x i . Here, m = i occurs with probability Ω i . Decoding is similarly performed in two separate steps. First the LT code is decoded, and then the outer code is decoded in the second step. Assuming that the outer code can recover the whole information block from any subset of n(R + σ),σ> 0 recovered intermediate bits, we focus our discussion on the LT decoder. For LT decoding, a decoding graph [2] is formed based on the set of received symbols. The decoding graph is a bipartite graph with one vertex set corresponding to the set of all intermediate bits, and the other set corresponding to the output bits (output nodes). Initially, each output node is adjacent to the group of intermediate nodes forming the corresponding received bits. Various decoding solutions can be used. Gaussian elimi- nation, although optimal, is typically too complex. A modi- fied version of the belief propagation algorithm, called edge deletion decoding (EDD) [1], is an efficient alternative when an appropriate design of Ω(x) is performed. EDD requires a small overhead in the number of received symbols for successful decoding [12]. This algorithm uses degree-one output nodes in the decoding graph to deduce the value of their neighbouring intermediate nodes, and then removes 0090-6778/12$31.00 c  2012 IEEE "©2012 IEEE. Personal use of this material is permitted. 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