Capacity achieving random sparse linear codes

2012 IEEE International Symposium on Information Theory Proceedings, 2012

In this paper the existence of capacity achieving linear codes with arbitrarily sparse generator matrices is proved. In particular, we show the existence of capacity achieving codes for which the density of ones in the generator matrix is arbitrarily low. The existing results on the existence of capacity achieving linear codes in the literature are limited to the codes whose generator matrix elements are zero or one with necessarily equal probability, yielding a non-sparse generator matrix. This will imply a high encoding complexity. An interesting trade-off between the sparsity of the generator matrix and the value of the error exponent is also demonstrated. Compared to the existing results in the literature, which are limited to codes with nonsparse generator matrices, the proposed approach is novel and more concise. Although the focus in this paper is on the Binary Symmetric and Binary Erasure Channels, the results can be easily extended to other discrete memoryless symmetric channels.

Information Theory, 2009. ISIT 2009. IEEE …, 2009

We establish a general framework for construction of small ensembles of capacity achieving linear codes for a wide range of (not necessarily memoryless) discrete symmetric channels, and in particular, the binary erasure and symmetric channels. The main tool used in our constructions is the notion of randomness extractors and lossless condensers that are regarded as central tools in theoretical computer science. Same as random codes, the resulting ensembles preserve their capacity achieving properties under any change of basis. Using known explicit constructions of condensers, we obtain specific ensembles whose size is as small as polynomial in the block length. By applying our construction to Justesen's concatenation scheme we obtain explicit capacity achieving codes for BEC (resp., BSC) with almost linear time encoding and almost linear time (resp., quadratic time) decoding and exponentially small error probability.

2009

We establish a general framework for construction of small ensembles of capacity achieving linear codes for a wide range of (not necessarily memoryless) discrete symmetric channels, and in particular, the binary erasure and symmetric channels. The main tool used in our constructions is the notion of randomness extractors and lossless condensers that are regarded as central tools in theoretical computer science. Same as random codes, the resulting ensembles preserve their capacity achieving properties under any change of basis. Using known explicit constructions of condensers, we obtain specific ensembles whose size is as small as polynomial in the block length. By applying our construction to Justesen's concatenation scheme (Justesen, 1972) we obtain explicit capacity achieving codes for BEC (resp., BSC) with almost linear time encoding and almost linear time (resp., quadratic time) decoding and exponentially small error probability.

arXiv (Cornell University), 2016

Random linear network coding (RLNC) in theory achieves the max-flow capacity of multicast networks, at the cost of high decoding complexity. To improve the performance-complexity tradeoff, we consider the design of sparse network codes. A generation-based strategy is employed in which source packets are grouped into overlapping subsets called generations. RLNC is performed only amongst packets belonging to the same generation throughout the network so that sparseness can be maintained. In this paper, generation-based network codes with low reception overheads and decoding costs are designed for transmitting of the order of 10 2-10 3 source packets. A low-complexity overhead-optimized decoder is proposed that exploits "overlaps" between generations. The sparseness of the codes is exploited through local processing and multiple rounds of pivoting of the decoding matrix. To demonstrate the efficacy of our approach, codes comprising a binary precode, random overlapping generations, and binary RLNC are designed. The results show that our designs can achieve negligible code overheads at low decoding costs, and outperform existing network codes that use the generation based strategy.

1999

IEEE Access, 2017

While random linear network coding is known to improve network reliability and throughput, its high costs for delivering coding coefficients and decoding represent an obstacle where nodes have limited power to transmit and decode packets. In this paper, we propose sparse network codes for scenarios where low coding vector weights and low decoding cost are crucial. We consider generation-based network codes where source packets are grouped into overlapping subsets called generations, and coding is performed only on packets within the same generation in order to achieve sparseness and low complexity. A sparse code is proposed that is comprised of a precode and random overlapping generations. The code is shown to be much sparser than existing codes that enjoy similar code overhead. To efficiently decode the proposed code, a novel low-complexity overhead-optimized decoder is proposed where code sparsity is exploited through local processing and multiple rounds of pivoting. Through extensive simulation comparison with existing schemes, we show that short transmissions of the order of 10 2 − 10 3 source packets, a denomination convenient for many applications of interest, can be efficiently decoded by the proposed decoder. INDEX TERMS Network coding, sparse codes, random codes, generations, code overhead, efficient decoding.

IEEE Transactions on Information Theory, 2000

This paper introduces ensembles of systematic accumulate-repeat-accumulate (ARA) codes which asymptotically achieve capacity on the binary erasure channel (BEC) with bounded complexity, per information bit, of encoding and decoding. It also introduces symmetry properties which play a central role in the construction of new capacity-achieving ensembles for the BEC. The results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity-achieving code ensembles defined on graphs. The superiority of ARA codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity-achieving ensembles of low-density parity-check (LDPC) and irregular repeat-accumulate (IRA) codes. ARA codes also have the advantage of being systematic.

2010

In this paper, new sequences (n ; n) of capacity achieving low-density parity-check (LDPC) code ensembles over the binary erasure channel (BEC) is introduced. These sequences include the existing sequences by Shokrollahi et al. as a special case. For a fixed code rate R, in the set of proposed sequences, Shokrollahi's sequences are superior to the rest of the set in that for any given value of n, their threshold is closer to the capacity upper bound 1 0 R. For any given , 0 < < 1 0 R, however, there are infinitely many sequences in the set that are superior to Shokrollahi's sequences in that for each of them, there exists an integer number n0, such that for any n > n0, the sequence (n ; n) requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within-neighborhood of the capacity upper bound 1 0 R. Moreover, it is proven that the check-regular subset of the proposed sequences are asymptotically quasi-optimal, i.e., their decoding complexity increases only logarithmically with the relative increase of the threshold. A stronger result on asymptotic optimality of some of the proposed sequences is also established. Index Terms-Asymptotically optimal sequences, binary erasure channel (BEC), capacity achieving sequences, check regular ensembles, low-density parity-check codes (LDPC).

2010

The linear, binary, block codes with no equally likely probabilities for the binary symbols are analyzed. The encoding graph for systematic linear block codes is proposed. These codes are seen as sources with memory and the information quantities H(S,X), H(S), H(X), H(X|S), H(S|X), I(S,X) are derived. On the base of these quantities, the code performances are analyzed.

2005

The paper introduces ensembles of accumulate-repeat-accumulate (ARA) codes which asymptotically achieve capacity on the binary erasure channel (BEC) with bounded complexity, per information bit, of encoding and decoding. It also introduces symmetry properties which play a central role in the construction of capacity-achieving ensembles for the BEC with bounded complexity. The results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity-achieving ensembles of codes defined on graphs. The superiority of ARA codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity-achieving ensembles of LDPC and IRA codes. The ARA codes also have the advantage of being systematic.