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Design of LDPC Lattice Network Codes Based on Construction D’

Paulo Branco Danilo Silva

Communications Research Group Electrical Engineering Department Federal University of Santa Catarina

SPCoding School Campinas, January 26, 2015

Paulo Branco, Danilo Silva Design of LDPC Lattice Network Codes Based on Construction D’ 1 / 5

Introduction Motivation

Multilevel codes, such as by Construction D, have high throughput, but code nesting makes the design of highly efficient codes difficult; Construction D’ offers high spectral efficiency and a design of codes based on matrix H, allowing for the design of efficient LDPC codes;

Problem

Messages wℓ ∈ W encoded to xℓ ∈ Cn. Decoding from y of the linear combination of the messages: D(y) = ˆ u = a1w1 + a2w2 + . . . + aℓwℓ a = (a1, . . . , aL) is an integer coefficient vector.

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Optimization Problem for Nested Codes

We have H of a linear [n, k] code, with R = k/n and m = n − k. Finding degree distributions λ = (λ2, . . . , λdℓ) and ρ = (ρ2, . . . , ρdr) maximizing the threshold (DE) is a known optimization problem subject to constraints; For nested codes the e.p. λ and ρ undergo new inequality

• constraints. We show only the case where C2 is nested in C1,

and we convert λ and ρ to n.p. α and β.

β(2)

i

≥ m(1) m(2) β(1)

i

, i = 2, . . . , d(1)

r d(2)

ℓ

• i=j

α(2)

i

≥

d(1)

ℓ

• i=j

α(1)

k ,

j = 2, . . . , d(1)

ℓ

Optimization problem: Maximize threshold(λ,ρ) s.t. inequality constraints on λ and ρ

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Signal Mapping to Lattice Points

1 Construction D’

Let C1 ⊆ C2 ⊆ . . . ⊆ CL−1 ⊆ CL be a family of nested binary linear codes. Let h1, h2, . . . , hn be a basis for Fn

2 such

that Ci is defined by the parity-check vectors h1, h2, . . . , hri. Let ΛD′ be the lattice consisting of all vectors x ∈ Zn satisfying the congruences: xT˜ σ (hj) ≡ 0 (mod 2i)

2 Linear Signal Mapping to Lattice Points

We define Hi, for levels i = 1, 2, . . . , L, as the parity-check matrices, and Gi as the respective generating matrices. We also define ui as the binary message for level i, w as the message in Z2L, and x as the lattice coded message.

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Linear Signal Mapping to Lattice Points

To obtain the linear mapping rule we use the following sequences

• f steps:

we code a series of canonical vectors vj, i.e., the basis vectors for the row space of Hi, where i = 1, 2, . . . , L, j = N − Ki−1 + 1, . . . N − Ki, and K0 = N, according to the following steps:

according to the level i = 1, 2, . . . , L, canonical vectors are multiplied by 2i−1; modifiy Hi to echelon form, obtaining Hechi; calculate di =

• piHT

echi/2i−1

mod 2, where p1 = 0; calculate si = [0 di]; pi+1 =

i

• k=1

2k−1 (gk,j + sk) mod 2, gi,j being the jth row of Gi, gi,j = vj ∗ Gi; The coded vj is wj = pL+1/2i−1, where i indicates the first level i for which vj ∈ Hi;

we stack all coded canonical vectors wj from all levels i in a matrix, yielding the lattice generator matrix G; linear message coding becomes x = (w ∗ G) mod 2L.

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