Finite-Length Analysis of Irregular Expurgated LDPC Codes under ...

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Asymptotic Expansion Asymptotic Expansion w.r.t n while t is fixed P b (n, ǫ, t) = P b (∞, ǫ, t) + α(ǫ, t) 1 n + O ( 1 n 2 ) Coefficient of 1/n α(ǫ, t) := lim n→∞ n (P b(n, ǫ, t) − P b (∞, ǫ, t)) Approximation P b (n, ǫ, t) ≈ P b (∞, ǫ, t) + α(ǫ, t) 1 n Our purpose is to derive α(ǫ, t) for irregular ensembles 4 / 15
Neighborhoods P b (n, ǫ, t) = ∑ P n (G)P b (G, ǫ) G ∈ the set of all neighborhoods of depth t P b (G, ǫ) ε(1−(1−ε) 2 ) 2 ε 2 (1−(1−ε) 2 ) ε 3 ε(1−(1−ε) 2 ) ε 2 (1+ε(1−ε)) ε P n (G) (2n-6)(2n-8) (2n-1)(2n-5) 2(2n-6) (2n-1)(2n-5) 1 (2n-1)(2n-5) 2 (2n-1)(2n-5) 4(2n-6) (2n-1)(2n-5) 2 (2n-1) Order of P n (G) -1 -2 -2 -1 -1 1 n n n n n G 5 / 15
Page 1 and 2: Finite-Length Analysis of Irregular
Page 3: Previous Results Analysis for the B
Page 7 and 8: Calculation of α(ǫ, t) α(ǫ, t)
Page 9 and 10: Method of Generating Function E t [
Page 11 and 12: α(ǫ, t) for Optimized Irregular E
Page 13 and 14: Ensembles with λ 2 = 0 10 1 0.15 0
Page 15: Conclusion and Open Problems Conclu

Finite-Length Analysis of Irregular Expurgated LDPC Codes under ...

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