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.
Rateless codes such as LT and Raptor codes, can be considered as a binary implementation of SNC @cite_18 @cite_0 and partial decoding probability has been a major research topic in the literature of rateless codes. To mention a few, the authors of @cite_14 designed an algorithm for an optimal recovery rate, i.e. partial decoding probability in LT codes. However, the results of this work are asymptotically optimal and may only be employed for infinite number of source packets. @cite_17 , the authors provided a probability analysis for decoding a fraction of source packets based on the structure of the received coded packets. However, their analysis can not provide any probability for partial decoding, where the exact structure of the received coded packets are unknown. The authors of @cite_9 and @cite_28 , proposed algorithms to increase the probability of partial decoding for rateless codes. However, these works only increase the probability of partial decoding in the specific stages of the whole transmission and also can not be extended to non-binary Galois Fields. Also these algorithms are based on current coded packets received by the decoder and require huge computation overhead while transmitting coded packets.
@cite_18 @cite_14 @cite_28 @cite_9 @cite_0 @cite_17
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In this correspondence, a generalization of rateless codes is proposed. The proposed codes provide unequal error protection (UEP). The asymptotic properties of these codes under the iterative decoding are investigated. Moreover, upper and lower bounds on maximum-likelihood (ML) decoding error probabilities of finite-length LT and Raptor codes for both equal and unequal error protection schemes are derived. Further, our work is verified with simulations. Simulation results indicate that the proposed codes provide desirable UEP. We also note that the UEP property does not impose a considerable drawback on the overall performance of the codes. Moreover, we discuss that the proposed codes can provide unequal recovery time (URT). This means that given a target bit error rate, different parts of information bits can be decoded after receiving different amounts of encoded bits. This implies that the information bits can be recovered in a progressive manner. This URT property may be used for sequential data recovery in video audio streaming The performance of a novel fountain coding scheme based on maximum distance separable (MDS) codes constructed over Galois fields of order q>=2 is investigated. Upper and lower bounds on the decoding failure probability under maximum likelihood decoding are developed. Differently from Raptor codes (which are based on a serial concatenation of a high-rate outer block code, and an inner Luby-transform code), the proposed coding scheme can be seen as a parallel concatenation of an outer MDS code and an inner random linear fountain code, both operating on the same Galois field. A performance assessment is performed on the gain provided by MDS based fountain coding over linear random fountain coding in terms of decoding failure probability vs. overhead. It is shown how, for example, the concatenation of a (15,10) Reed-Solomon code and a linear random fountain code over F16 brings to a decoding failure probability 4 orders of magnitude lower than the linear random fountain code for the same overhead in a channel with a packet loss probability of epsilon=0.05. Moreover, it is illustrated how the performance of the concatenated fountain code approaches that of an idealized fountain code for higher-order Galois fields and moderate packet loss probabilities. The scheme introduced is of special interest for the distribution of data using small block sizes. We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process that adds the errors can be described by a sufficiently “simple” circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder decoder that works for every channel in the class. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. Unique decoding for additive errors: We give the first construction of a polynomial-time encodable decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity 1 − H(p). These are channels that add an arbitrary error vector e ∈ 0, 1 N of weight at most pN to the transmitted word; the vector e can depend on the code but not on the randomness of the encoder or the particular transmitted word. Such channels capture binary symmetric errors and burst errors as special cases. List decoding for polynomial-time channels: For every constant c > 0, we construct codes with optimal rate (arbitrarily close to 1 − H(p)) that efficiently recover a short list containing the correct message with high probability for channels describable by circuits of size at most Nc. Our construction is not fully explicit but rather Monte Carlo (we give an algorithm that, with high probability, produces an encoder decoder pair that works for all time Nc channels). We are not aware of any channel models considered in the information theory literature other than purely adversarial channels, which require more than linear-size circuits to implement. We justify the relaxation to list decoding with an impossibility result showing that, in a large range of parameters (p > 1 4), codes that are uniquely decodable for a modest class of channels (online, memoryless, nonuniform channels) cannot have positive rate. In this paper, collocated and distributed space-time block codes (DSTBCs) which admit multi-group maximum likelihood (ML) decoding are studied. First the collocated case is considered and the problem of constructing space-time block codes (STBCs) which optimally tradeoff rate and ML decoding complexity is posed. Recently, sufficient conditions for multi-group ML decodability have been provided in the literature and codes meeting these sufficient conditions were called Clifford Unitary Weight (CUW) STBCs. An algebraic framework based on extended Clifford algebras is proposed to study CUW STBCs and using this framework, the optimal tradeoff between rate and ML decoding complexity of CUW STBCs is obtained for few specific cases. Code constructions meeting this tradeoff optimally are also provided. The paper then focuses on multi-group ML decodable DSTBCs for application in synchronous wireless relay networks and three constructions of four-group ML decodable DSTBCs are provided. Finally, the OFDM based Alamouti space-time coded scheme proposed by Li-Xia for a 2 relay asynchronous relay network is extended to a more general transmission scheme that can achieve full asynchronous cooperative diversity for arbitrary number of relays. It is then shown how differential encoding at the source can be combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full cooperative diversity in asynchronous wireless relay networks with no channel information and also no timing error knowledge at the destination node. Four-group decodable DSTBCs applicable in the proposed OFDM based transmission scheme are also given.