Sparse Network Coding (SNC) has been a promising network coding scheme as an improvement for Random Linear Network Coding (RLNC) in terms of the computational complexity. However, in this literature, there has been no analytical expressions for the probability of decoding a fraction of source messages after transmission of some coded packets. In this work, we looked into the problem of the probability of decoding a fraction of source messages, i.e., partial decoding, in the decoder for a system which uses SNC. We exploited the Principle of Inclusion and Exclusion to derive expressions of partial decoding probability. The presented model predicts the probability of partial decoding with an average deviation of 6 . Our results show that SNC has a great potential for recovering a fraction of the source message, especially in higher sparsity and lower Galois Field size. Moreover, to achieve a better probability of partial decoding throughout transmissions, we define a sparsity tuning scheme that significantly increases the probability of partial decoding. Our results show that this tuning scheme achieves a 16 improvement in terms of probability of decoding a fraction of source packets with respect to traditional SNC.
 Fountain codes are designed so that all input symbols can be recovered from a slightly larger number of coded symbols, with high probability using an iterative decoder. In this paper we investigate the number of input symbols that can be recovered by the same decoder, but when the number of coded symbols available is less than the total number of input symbols. Of course recovery of all inputs is not possible, and the fraction that can be recovered will depend on the output degree distribution of the code. In this paper we (a) outer bound the fraction of inputs that can be recovered for any output degree distribution of the code, and (b) design degree distributions which meet perform close to this bound. Our results are of interest for real-time systems using rateless codes, and for Raptor-type two-stage designs. Rateless erasure resilient codes, e.g., LT codes, can recover all message symbols perfectly with high probability from a slightly greater number of encoding symbols. However, it has been observed that only a few message symbols can be recovered from a less number of encoding symbols. In this paper, we investigate the performance of rateless codes when the number of received encoding symbols is less than that of the original message symbols. Through ordered generation of encoding symbols according to their degrees, we demonstrate that a significantly improved intermediate performance of rateless codes can be obtained. Existing rateless codes have low intermediate symbol recovery rates (ISRR). Therefore, we first design new rateless codes with close to optimal ISRR employing genetic algorithms. Next, we assume an estimate of the channel erasure rate is available and propose an algorithm to further improve the ISRR of the designed codes.  This paper provides an efficient method for analyzing the error probability of the belief propagation (BP) decoder applied to LT Codes. Each output symbol is generated independently by sampling from a distribution and adding the input symbols corresponding to the support of the sampled vector.