Coding Theorem for Systematic LDGM Codes Under List Decoding

Lin, Wenchao; Cai, Suihua; Wei, Baodian; Ma, Xiao

doi:10.1109/itw.2018.8613510

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…Furthermore, the scaling exponent of such random linear codes are studied in [16]. Later, in [17], the existence of capacity-achieving systematic LDGM ensembles over any BMS channel with the expected value of the weight of the entire generator matrix bounded by N 2 , for any > 0, is shown.…”

Section: A Ldgm and Related Workmentioning

confidence: 99%

“…Thus, for any fixed , λ( , α) is a concave function of α and has maximum when ∂λ( , α) ∂α = 0.From (42), the above equality holds if and only if α = 1 6 − and attains the maximum value when max λ( , α i ) max α λ( , α) = * − < 0 for any 0 i n − n lub − 1. Hence, by(17) and (41), γ, which is upper bounded by n(n + 1) 2 2 n( * − ) . = 2 n( * − ) , approaches 0 exponentially fast.…”

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confidence: 91%

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Capacity-achieving Polar-based LDGM Codes with Crowdsourcing Applications

Pang

Mahdavifar

Pradhan

2020

2020 IEEE International Symposium on Information Theory (ISIT)

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In this paper, we study codes with sparse generator matrices. More specifically, low-density generator matrix (LDGM) codes with a certain constraint on the weight of all the columns in the generator matrix are considered. Previous works have shown the existence of capacity-achieving LDGM codes with column weight bounded above by any linear function of the block length N over the BSC, and by the logarithm of N over the BEC. In this paper, it is first shown that when a binary-input memoryless symmetric (BMS) channel W and a constant s > 0 are given, there exists a polarization kernel such that the corresponding polar code is capacity-achieving and the column weights of the generator matrices are bounded from above by N s. Then, a general construction based on a concatenation of polar codes and a rate-1 code, and a new column-splitting algorithm that guarantees a much sparser generator matrix is given. More specifically, for any BMS channel and any > 2 * , where * ≈ 0.085, an existence of sequence of capacityachieving codes with all the column wights of the generator matrix upper bounded by (log N) 1+ is shown. Furthermore, coding schemes for BEC and BMS channels, based on a second column-splitting algorithm are devised with low-complexity decoding that uses successive-cancellation. The second splitting algorithm allows for the use of a low-complexity decoder by preserving the reliability of the bitchannels observed by the source bits, and by increasing the code block length. In particular, for any BEC and any λ > λ * = 0.5 + * , the existence of a sequence of capacity-achieving codes where all the column wights of the generator matrix are bounded from above by (log N) 2λ and with decoding complexity O(N log log N) is shown. The existence of similar capacity-achieving LDGM codes with low-complexity decoding is shown for any BMS channel, and for any λ > λ † ≈ 0.631.

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Section: A Ldgm and Related Workmentioning

confidence: 99%

mentioning

confidence: 91%

Capacity-achieving Polar-based LDGM Codes with Crowdsourcing Applications

Pang

Mahdavifar

Pradhan

2020

2020 IEEE International Symposium on Information Theory (ISIT)

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“…Furthermore, the scaling exponent of such random linear codes are studied in [15]. Later, in [16], the existence of capacity achieving systematic LDGM ensembles over any BMS channel with the expected value of the weight of the entire generator matrix bounded by ǫN 2 , for any ǫ > 0, is shown.…”

Section: A Ldgm and Related Workmentioning

confidence: 99%

“…)⌋ and the analysis still holds). By grouping the k max terms in (16), the ratio R can be expressed as a sum of log k max = n − n lub terms, as shown in the following lemma. ).…”

Section: E Sparsity With Kernel Gmentioning

confidence: 99%

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Capacity-achieving Polar-based LDGM Codes with Crowdsourcing Applications

Pang¹,

Mahdavifar²,

Pradhan³

2020

Preprint

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In this paper we study codes with sparse generator matrices. More specifically, codes with a certain constraint on the weight of all the columns in the generator matrix are considered. The end result is the following. For any binary-input memoryless symmetric (BMS) channel and any ǫ > 2ǫ * , where ǫ * = 1 6 − 5 3 log 4 3 ≈ 0.085, we show an explicit sequence of capacity-achieving codes with all the column wights of the generator matrix upper bounded by (log N ) 1+ǫ , where N is the code block length. The constructions are based on polar codes. Applications to crowdsourcing are also shown.

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