Design of LDPC Lattice Network Codes Based on Construction D Paulo - PowerPoint PPT Presentation

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 w ℓ ∈ W encoded to x ℓ ∈ C n . Decoding from y of the linear combination of the messages: D ( y ) = ˆ u = a 1 w 1 + a 2 w 2 + . . . + a ℓ w ℓ a = ( a 1 , . . . , a L ) is an integer coefficient vector. Paulo Branco, Danilo Silva Design of LDPC Lattice Network Codes Based on Construction D’ 2 / 5

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 = ( λ 2 , . . . , λ d ℓ ) and ρ = ( ρ 2 , . . . , ρ d r ) 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 C 2 is nested in C 1 , and we convert λ and ρ to n.p. α and β . ≥ m (1) β (2) m (2) β (1) i = 2 , . . . , d (1) , r i i d (2) d (1) ℓ ℓ � α (2) � α (1) j = 2 , . . . , d (1) k , ≥ i ℓ i = j i = j Optimization problem: Maximize threshold( λ , ρ ) s.t. inequality constraints on λ and ρ Paulo Branco, Danilo Silva Design of LDPC Lattice Network Codes Based on Construction D’ 3 / 5

Signal Mapping to Lattice Points 1 Construction D’ Let C 1 ⊆ C 2 ⊆ . . . ⊆ C L − 1 ⊆ C L be a family of nested binary linear codes. Let h 1 , h 2 , . . . , h n be a basis for F n 2 such that C i is defined by the parity-check vectors h 1 , h 2 , . . . , h r i . Let Λ D ′ be the lattice consisting of all vectors x ∈ Z n satisfying the congruences: x T ˜ (mod 2 i ) σ ( h j ) ≡ 0 2 Linear Signal Mapping to Lattice Points We define H i , for levels i = 1 , 2 , . . . , L , as the parity-check matrices, and G i as the respective generating matrices. We also define u i as the binary message for level i , w as the message in Z 2 L , and x as the lattice coded message. Paulo Branco, Danilo Silva Design of LDPC Lattice Network Codes Based on Construction D’ 4 / 5

Linear Signal Mapping to Lattice Points To obtain the linear mapping rule we use the following sequences of steps: we code a series of canonical vectors v j , i.e., the basis vectors for the row space of H i , where i = 1 , 2 , . . . , L , j = N − K i − 1 + 1 , . . . N − K i , and K 0 = N , according to the following steps: according to the level i = 1 , 2 , . . . , L , canonical vectors are multiplied by 2 i − 1 ; modifiy H i to echelon form, obtaining H ech i ; � p i H T ech i / 2 i − 1 � calculate d i = mod 2 , where p 1 = 0 ; calculate s i = [ 0 d i ]; i 2 k − 1 ( g k , j + s k ) mod 2 , g i , j being the j th row of � p i +1 = k =1 G i , g i , j = v j ∗ G i ; The coded v j is w j = p L +1 / 2 i − 1 , where i indicates the first level i for which v j ∈ H i ; we stack all coded canonical vectors w j from all levels i in a matrix, yielding the lattice generator matrix G ; linear message coding becomes x = ( w ∗ G ) mod 2 L . Paulo Branco, Danilo Silva Design of LDPC Lattice Network Codes Based on Construction D’ 5 / 5