Construction of Rate ( n 1) / n Non-Binary LDPC Convolutional Codes - PowerPoint PPT Presentation

Construction of Rate ( n − 1) / n Non-Binary LDPC Convolutional Codes via Difference Triangle Sets Gianira N. Alfarano, Julia Lieb, Joachim Rosenthal University of Zurich gianiranicoletta.alfarano@math.uzh.ch ISIT 2020: IEEE International Symposium on Information Theory

Preliminaries New construction Conclusions and future works Outline Preliminaries New construction Conclusions and future works

Preliminaries New construction Conclusions and future works Outline Preliminaries New construction Conclusions and future works

Preliminaries New construction Conclusions and future works Convolutional codes • F q finite field of q elements, q prime power • n , k positive integers • F q [ z ] polynomial ring over F q Definition An ( n , k ) q convolutional code is a F q [ z ]-submodule C ⊆ F q [ z ] n of rank k . Remark There exists a generator matrix G ( z ) ∈ F q [ z ] k × n of rank k such that C := { uG ( z ) | u ∈ F q [ z ] k } ⊆ F q [ z ] n . If G ( z ) is reduced and basic , there exists a parity-check matrix H ( z ) ∈ F q [ z ] ( n − k ) × n , such that C := { v ∈ F q [ z ] n | H ( z ) v ⊤ = 0 } = ker( H ( z )) .

Preliminaries New construction Conclusions and future works Convolutional codes • F q finite field of q elements, q prime power • n , k positive integers • F q [ z ] polynomial ring over F q Definition An ( n , k ) q convolutional code is a F q [ z ]-submodule C ⊆ F q [ z ] n of rank k . Remark There exists a generator matrix G ( z ) ∈ F q [ z ] k × n of rank k such that C := { uG ( z ) | u ∈ F q [ z ] k } ⊆ F q [ z ] n . If G ( z ) is reduced and basic , there exists a parity-check matrix H ( z ) ∈ F q [ z ] ( n − k ) × n , such that C := { v ∈ F q [ z ] n | H ( z ) v ⊤ = 0 } = ker( H ( z )) .

Preliminaries New construction Conclusions and future works Minimal and basic matrices Definition Let G ( z ) ∈ F q [ z ] k × n of rank k . • G(z) is basic if its Smith-form is given by � � 0 . I k • G(z) is reduced if the sum of its row degrees attains the minimal possible value Definition The degree δ of the code is the largest degree of the k × k minors of G ( z ). Notation We denote an ( n , k ) q convolutional code with degree δ by ( n , k , δ ) q convolutional code.

Preliminaries New construction Conclusions and future works Parameters • F q [ z ] n ∼ = F n q [ z ] i u i z i �→ � • wt : F n q [ z ] → N , � i wt ( u i ) • d free := min v ( z ) � =0 { wt ( v ( z )) | v ( z ) ∈ C} Generalized Singleton bound �� δ � � d free ≤ ( n − k ) + 1 + δ + 1 k

Preliminaries New construction Conclusions and future works Parameters • F q [ z ] n ∼ = F n q [ z ] i u i z i �→ � • wt : F n q [ z ] → N , � i wt ( u i ) • d free := min v ( z ) � =0 { wt ( v ( z )) | v ( z ) ∈ C} Generalized Singleton bound �� δ � � d free ≤ ( n − k ) + 1 + δ + 1 k Remark • For δ = 0 the upper bound reduces to block codes situation. • If d free ( C ) reaches the bound, C is said to be MDS .

Preliminaries New construction Conclusions and future works Parameters • F q [ z ] n ∼ = F n q [ z ] i u i z i �→ � • wt : F n q [ z ] → N , � i wt ( u i ) • d free := min v ( z ) � =0 { wt ( v ( z )) | v ( z ) ∈ C} Generalized Singleton bound �� δ � � d free ≤ ( n − k ) + 1 + δ + 1 k Remark • For δ = 0 the upper bound reduces to block codes situation. • If d free ( C ) reaches the bound, C is said to be MDS .

Preliminaries New construction Conclusions and future works Sliding parity-check matrix Let C be an ( n , k , δ ) q convolutional code. The isomorphism F q [ z ] n → F n q [ z ] allows to consider H ( z ) ∈ F ( n − k ) × n [ z ], such that q H ( z ) = H 0 + H 1 z + . . . H µ z µ , with µ > 0. − → expand the kernel representation   H 0 . ... .   .     v 0   · · · H µ H 0   v 1 Hv ⊤ =  ... ...     = 0 ,    .  .     .    H µ · · · H 0   v r .  ...  .   .   H µ with r = ⌊ δ k ⌋ .

Preliminaries New construction Conclusions and future works j-th column distances For any j ∈ N 0 we define the j -th column distance of C as � � d c wt ( v 0 + v 1 z + · · · + v j z j ) | v ( z ) ∈ C , v 0 � = 0 j ( C ) := min � � j [ v 0 · · · v j ] ⊤ = 0 wt ( v 0 + · · · + v j z j ) | v 0 � = 0 , H c = min   H 0 H 1 H 0   H c j :=  . .  ... . .   . .   H j H j − 1 · · · H 0 .

Preliminaries New construction Conclusions and future works Non-binary LDPC convolutional codes Definition A NB-LDPC convolutional code is defined as the kernel of a sparse sliding parity-check matrix H . • We can associate to H a bipartite graph G = ( V , E ), called Tanner graph , where V = V s ∪ V c is the set of vertices. • V s is the set of the row indexes and V c is the set of column indexes. • E ⊆ V s × V c is the set of edges, with e i , j = ( v i , c j ) ∈ E if and only if h i , j � = 0. • For an even integer ℓ , we call a simple closed path consisting of ℓ/ 2 check nodes and ℓ/ 2 variable nodes in G an ℓ -cycle .

Preliminaries New construction Conclusions and future works Full rank condition Theorem An ℓ -cycle in a NB-LDPC code with parity check matrix H can be represented by an ℓ 2 × ℓ 2 submatrix of H of the form   a 1 a 2 0 · · · · · · 0 . .   0 a 3 a 4 · · · · · · .   . .  ...  . .   . . A = ,   . .  ...  . .  . .     0  a ℓ − 3 a ℓ − 2   a ℓ 0 · · · · · · 0 a ℓ − 1 where a i ∈ F ∗ q . The cycle satisfy the FRC if det( A ) � = 0 . − → Avoid short cycles not satisfying the FRC.

Preliminaries New construction Conclusions and future works Outline Preliminaries New construction Conclusions and future works

Preliminaries New construction Conclusions and future works Difference Triangle Sets Definition (( N , M )- difference triangle set (DTS)) • T := { T 1 , T 2 , . . . , T N } , where T i ⊂ N 0 and | T i | = M . • T i := { a i , j | 1 ≤ j ≤ M } , with a i , 1 < a i , 2 < · · · < a i , M • a i , j − a i , k , with 1 ≤ i ≤ N and 1 ≤ k < j ≤ M distinct . • m ( T ) := max { a i , M | 1 ≤ i ≤ N } is called scope . Example T = {{ 0 , 3 , 15 , 19 } , { 0 , 6 , 11 , 13 } , { 0 , 8 , 17 , 18 }} is a (3 , 4) DTS with scope m ( T ) = 19.

Preliminaries New construction Conclusions and future works Convolutional Codes from DTS Remark • The decoding of an ( n , k , δ ) convolutional code C is done sequentially by blocks of length n . • The error-correcting properties of the code are determined by the decoding of the first block of the sliding parity-check matrix   H 0 H 1 H 0   B := H c µ =  . .  ... . .   . .   · · · H µ H µ − 1 H 0

Preliminaries New construction Conclusions and future works Convolutional Codes from DTS Remark • The decoding of an ( n , k , δ ) convolutional code C is done sequentially by blocks of length n . • The error-correcting properties of the code are determined by the decoding of the first block of the sliding parity-check matrix   H 0 H 1 H 0   B := H c µ =  . .  ... . .   . .   · · · H µ H µ − 1 H 0

Preliminaries New construction Conclusions and future works Convolutional Codes from DTS Remark • The decoding of an ( n , k , δ ) convolutional code C is done sequentially by blocks of length n . • The error-correcting properties of the code are determined by the decoding of the first block of the sliding parity-check matrix   | A 0 I n − k A 1 | 0 H 0   B := H c µ =  . . .  ... . . .   . | . .   | 0 · · · A µ H µ − 1 H 0

Preliminaries New construction Conclusions and future works Convolutional Codes from DTS Remark • The decoding of an ( n , k , δ ) convolutional code C is done sequentially by blocks of length n . • The error-correcting properties of the code are determined by the decoding of the first block of the sliding parity-check matrix   H 0   B := H c M µ = .  ...  .   .   H µ − 1 · · · H 0

Preliminaries New construction Conclusions and future works Summarizing Remark • Construct the sliding parity-check matrix H of a NB-LDPC such that the Tanner graph G associated to H does not contain short cycles not satisfying the FRC. • H satisfies this property if and only if B does. • Construct B , such that all the 2 × 2 and 3 × 3 minors that are non-trivially zero, are non-zero to avoid 4 and 6 cycles not satisfying the FRC. • We focus on the case rate n − 1 / n

Preliminaries New construction Conclusions and future works Construction Definition (Construction) 1. Let T := { T 1 , . . . , T n − 1 } be an ( n − 1 , w )-DTS. 2. Let α be a primitive element for F q . � α ik if i ∈ T k 3. For any 1 ≤ i ≤ m ( T ), 1 ≤ k ≤ n − 1, define M T i , k = otherwise . 0 4. Set the column M T i , n equal to [1 , 0 , · · · , 0] ⊤ . 5. Derive a sliding matrix H T by “shifting” the columns of M T . 6. Define C T := ker( H T ) over F q .