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

• Fq finite field of q elements, q prime power
• n, k positive integers
• Fq[z] polynomial ring over Fq

Definition

An (n, k)q convolutional code is a Fq[z]-submodule C ⊆ Fq[z]n of rank k.

Remark

There exists a generator matrix G(z) ∈ Fq[z]k×n of rank k such that C := {uG(z) | u ∈ Fq[z]k} ⊆ Fq[z]n. If G(z) is reduced and basic, there exists a parity-check matrix H(z) ∈ Fq[z](n−k)×n, such that C := {v ∈ Fq[z]n | H(z)v⊤ = 0} = ker(H(z)).

Preliminaries New construction Conclusions and future works

Convolutional codes

• Fq finite field of q elements, q prime power
• n, k positive integers
• Fq[z] polynomial ring over Fq

Definition

An (n, k)q convolutional code is a Fq[z]-submodule C ⊆ Fq[z]n of rank k.

Remark

There exists a generator matrix G(z) ∈ Fq[z]k×n of rank k such that C := {uG(z) | u ∈ Fq[z]k} ⊆ Fq[z]n. If G(z) is reduced and basic, there exists a parity-check matrix H(z) ∈ Fq[z](n−k)×n, such that C := {v ∈ Fq[z]n | H(z)v⊤ = 0} = ker(H(z)).

Preliminaries New construction Conclusions and future works

Minimal and basic matrices

Definition

Let G(z) ∈ Fq[z]k×n of rank k.

• G(z) is basic if its Smith-form is given by
• Ik
• .
• 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

• Fq[z]n ∼

= Fn

q[z]

• wt : Fn

q[z] → N,

• i uizi →

i wt(ui)

• dfree := minv(z)=0{wt(v(z)) | v(z) ∈ C}

Generalized Singleton bound

dfree ≤ (n − k) δ k

• + 1
• + δ + 1

Preliminaries New construction Conclusions and future works

Parameters

• Fq[z]n ∼

= Fn

q[z]

• wt : Fn

q[z] → N,

• i uizi →

i wt(ui)

• dfree := minv(z)=0{wt(v(z)) | v(z) ∈ C}

Generalized Singleton bound

dfree ≤ (n − k) δ k

• + 1
• + δ + 1

Remark

• For δ = 0 the upper bound reduces to block codes situation.
• If dfree(C) reaches the bound, C is said to be MDS.

Preliminaries New construction Conclusions and future works

Parameters

• Fq[z]n ∼

= Fn

q[z]

• wt : Fn

q[z] → N,

• i uizi →

i wt(ui)

• dfree := minv(z)=0{wt(v(z)) | v(z) ∈ C}

Generalized Singleton bound

dfree ≤ (n − k) δ k

• + 1
• + δ + 1

Remark

• For δ = 0 the upper bound reduces to block codes situation.
• If dfree(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 Fq[z]n → Fn

q[z] allows to consider H(z) ∈ F(n−k)×n q

[z], such that H(z) = H0 + H1z + . . . Hµzµ, with µ > 0. − → expand the kernel representation Hv⊤ =              H0 . . . ... Hµ · · · H0 ... ... Hµ · · · H0 ... . . . Hµ                   v0 v1 . . . vr      = 0, with r = ⌊ δ

k ⌋.

Preliminaries New construction Conclusions and future works

j-th column distances

For any j ∈ N0 we define the j-th column distance of C as dc

j (C) := min

• wt(v0 + v1z + · · · + vjzj) | v(z) ∈ C, v0 = 0
• = min
• wt(v0 + · · · + vjzj) | v0 = 0,

Hc

j [v0 · · · vj]⊤ = 0

• Hc

j :=

     H0 H1 H0 . . . . . . ... Hj Hj−1 · · · H0      .

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 = Vs ∪ Vc is the set of vertices.

• Vs is the set of the row indexes and Vc is the set of column indexes.
• E ⊆ Vs × Vc is the set of edges, with ei,j = (vi, cj) ∈ E if and only if hi,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 =            a1 a2 · · · · · · a3 a4 · · · · · · . . . . . . ... . . . . . . ... . . . aℓ−3 aℓ−2 aℓ · · · · · · aℓ−1            , where ai ∈ 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 := {T1, T2, . . . , TN}, where Ti ⊂ N0 and |Ti| = M.
• Ti := {ai,j | 1 ≤ j ≤ M}, with ai,1 < ai,2 < · · · < ai,M
• ai,j − ai,k, with 1 ≤ i ≤ N and 1 ≤ k < j ≤ M distinct.
• m(T ) := max{ai,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 B := Hc

µ =

     H0 H1 H0 . . . . . . ... Hµ Hµ−1 · · · H0     

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 B := Hc

µ =

     H0 H1 H0 . . . . . . ... Hµ Hµ−1 · · · H0     

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 B := Hc

µ =

     A0 | In−k A1 | H0 . . . | . . . . . . ... Aµ | Hµ−1 · · · H0     

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 B := Hc

µ =

     M H0 . . . ... Hµ−1 · · · H0     

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 := {T1, . . . , Tn−1} be an (n − 1, w)-DTS.
• 2. Let α be a primitive element for Fq.
• 3. For any 1 ≤ i ≤ m(T ), 1 ≤ k ≤ n − 1, define MT

i,k =

• αik

if i ∈ Tk

• therwise .
• 4. Set the column MT

i,n equal to [1, 0, · · · , 0]⊤.

• 5. Derive a sliding matrix HT by “shifting” the columns of MT .
• 6. Define CT := ker(HT ) over Fq.

Preliminaries New construction Conclusions and future works

Example

Let Fq := {0, 1, α, . . . , αq−2} and T be a (2, 3)-DTS, such that T1 := {1, 2, 6} and T2 := {1, 2, 4}. Then, MT =         α α2 1 α2 α4 α8 α6        

Preliminaries New construction Conclusions and future works

Example

Let Fq := {0, 1, α, . . . , αq−2} and T be a (2, 3)-DTS, such that T1 := {1, 2, 6} and T2 := {1, 2, 4}. Then, MT =         α α2 1 α2 α4 α8 α6        

Preliminaries New construction Conclusions and future works

Example

Let Fq := {0, 1, α, . . . , αq−2} and T be a (2, 3)-DTS, such that T1 := {1, 2, 6} and T2 := {1, 2, 4}. Then, MT =         α α2 1 α2 α4 α8 α6        

Preliminaries New construction Conclusions and future works

Example

Let Fq := {0, 1, α, . . . , αq−2} and T be a (2, 3)-DTS, such that T1 := {1, 2, 6} and T2 := {1, 2, 4}. Then, MT =         α α2 1 α2 α4 α8 α6        

Preliminaries New construction Conclusions and future works

Example

Let Fq := {0, 1, α, . . . , αq−2} and T be a (2, 3)-DTS, such that T1 := {1, 2, 6} and T2 := {1, 2, 4}. Then, MT =         α α2 1 α2 α4 α8 α6        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1        

Preliminaries New construction Conclusions and future works

Example

The sliding parity-check matrix derived from MT is BT =         α α2 1 α2 α4 α α2 1 α2 α4 α α2 1 α8 α2 α4 α α2 1 α8 α2 α4 α α2 1 α6 α8 α2 α4 α α2 1         The code CT is a (3, 2, 5)q convolutional code.

Preliminaries New construction Conclusions and future works

Results

Proposition

Let T be an (n − 1, w)-DTS with scope m(T ). Then, the code CT is an (n, n − 1, m(T ) − 1)q convolutional code.

Theorem

Let T be an (n − 1, w)-DTS and consider the matrix

• A⊤

· · · A⊤

µ

⊤ defined as in the previous construction. Denote by wj the minimal column weight of A⊤ · · · A⊤

j

⊤. Then (i) dfree(CT ) = w + 1, (ii) dc

j (CT ) = wj + 1.

Remark

dc

j = dfree(CT ) for j ≥ µ.

Preliminaries New construction Conclusions and future works

Results

Proposition

Let T be an (n − 1, w)-DTS with scope m(T ). Then, the code CT is an (n, n − 1, m(T ) − 1)q convolutional code.

Theorem

Let T be an (n − 1, w)-DTS and consider the matrix

• A⊤

· · · A⊤

µ

⊤ defined as in the previous construction. Denote by wj the minimal column weight of A⊤ · · · A⊤

j

⊤. Then (i) dfree(CT ) = w + 1, (ii) dc

j (CT ) = wj + 1.

Remark

dc

j = dfree(CT ) for j ≥ µ.

Preliminaries New construction Conclusions and future works

Results

Let T be an (n − 1, w)-DTS with scope m(T ). Let CT be as in the Construction, defined over Fq.

Theorem

• If q > (n − 1)(m(T ) − 1) + 1, all the 4-cycles in the Tanner graph G satisfy the

FRC.

Theorem

• Let w ≥ 3.
• If, q = pN, where N > (m(T ) − 2)(n − 2), all the 6-cycles in the Tanner graph G

satisfy the FRC.

Preliminaries New construction Conclusions and future works

Outline

Preliminaries New construction Conclusions and future works

Preliminaries New construction Conclusions and future works

Conclusions and future works

• We gave a construction of rate (n − 1)/n LDPC convolutional codes over

non-binary fields, using difference triangle sets.

• We related the important parameters of the code to the parameters of the

considered DTS.

• We can ensure that the 4 and 6-cycles in the Tanner graph associated to codes

CT defined over q = pN, with N > (m(T ) − 2)(n − 2) satisfy the FRC.

• Minors of BT of larger size than 3 × 3 could be considered to derive convolutional

codes with larger distances. Unfortunately, this may require a larger field size.

• Last results can be generalized to arbitrary k. However, it is not completely trivial

anymore to determine the degree δ with the help of the parity-check matrix of the code.

Preliminaries New construction Conclusions and future works

Thank You!