Convolutional Code Construction from BlockCodes over the Galois Ring GR(pr ,m)

A Basic Research funded by OVCRE

Bernice Ruth P. CerezoCharles R. Repizo

Marjo-Anne B. Acob

March 2008 - February 2009

Project Background

Virgilio P. Sison, “Convolutional codes from linear block codesover Galois rings”

Bernice Ruth P. Cerezo, “Binary Convolutional Codes andLinear Block Codes over F4”

IMSP Coding Theory Research Cluster

Project Background

Virgilio P. Sison, “Convolutional codes from linear block codesover Galois rings”

Bernice Ruth P. Cerezo, “Binary Convolutional Codes andLinear Block Codes over F4”

IMSP Coding Theory Research Cluster

Project Background

Virgilio P. Sison, “Convolutional codes from linear block codesover Galois rings”

Bernice Ruth P. Cerezo, “Binary Convolutional Codes andLinear Block Codes over F4”

IMSP Coding Theory Research Cluster

Objectives

1. To extend the construction of Massey, Costello and Justesento codes over the Galois ring GR(pr ,m) and generalize thebound for the free distance of the convolutional code withrespect to the homogeneous weight;

2. To sharpen the bound given in the C (B, h,G ) construction byusing the image of the block code over GR(pr ,m) withrespect to some basis of GR(pr ,m)

Objectives

1. To extend the construction of Massey, Costello and Justesento codes over the Galois ring GR(pr ,m) and generalize thebound for the free distance of the convolutional code withrespect to the homogeneous weight;

2. To sharpen the bound given in the C (B, h,G ) construction byusing the image of the block code over GR(pr ,m) withrespect to some basis of GR(pr ,m)

Objectives

3. To extend the construction of Sidorenko, Medina and Bossertto codes over Galois rings, and prove a bound for thehomogeneous free distance of the convolutional code;

4. To apply these generalized constructions to certain knownblock codes over GR(pr ,m)

Objectives

3. To extend the construction of Sidorenko, Medina and Bossertto codes over Galois rings, and prove a bound for thehomogeneous free distance of the convolutional code;

4. To apply these generalized constructions to certain knownblock codes over GR(pr ,m)

Objective 1

Massey, Costello and Justesen, “Polynomial Weights andCode Constructions”

Justesen, ”New Convolutional Code Constructions and a Classof Asymptotically Good Time-Varying Codes“

The construction used in these two papers may be extendedto construct convolutional codes from cyclic codes overGR(4,m) and also GR(pr ,m).

Objective 1

Massey, Costello and Justesen, “Polynomial Weights andCode Constructions”

Justesen, ”New Convolutional Code Constructions and a Classof Asymptotically Good Time-Varying Codes“

The construction used in these two papers may be extendedto construct convolutional codes from cyclic codes overGR(4,m) and also GR(pr ,m).

Objective 1

Massey, Costello and Justesen, “Polynomial Weights andCode Constructions”

Justesen, ”New Convolutional Code Constructions and a Classof Asymptotically Good Time-Varying Codes“

The construction used in these two papers may be extendedto construct convolutional codes from cyclic codes overGR(4,m) and also GR(pr ,m).

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Construction

Consider the Galois Ring R := GR(4,m).

Let C be a cyclic code of odd length n over R.

C is generated by a polynomial g(x) which divides xn − 1.

Using the division algorithm, express g(x) as2l−1∑j=0

x jGj(x2l)

Convolutional code over Z4 of rate 1/2l that is generated byG (D) = g(D)

Polynomial generator matrix:[G0(D) G1(D) . . . G2l−1(D)

]

Objective 3

Sidorenko, et al, “From Block to Convolutional Codes usingBlock Distances”

Convolutional code C constructed from a block code over R

dfree(C) ≥ dB(C)

Objective 3

Sidorenko, et al, “From Block to Convolutional Codes usingBlock Distances”

Convolutional code C constructed from a block code over R

dfree(C) ≥ dB(C)

Objective 3

Sidorenko, et al, “From Block to Convolutional Codes usingBlock Distances”

Convolutional code C constructed from a block code over R

dfree(C) ≥ dB(C)

Objective 2

C (B, h,G ) construction by Sole and Sison

Let C be a block code over R with polynomial generatormatrix G with minimum distance d

Each entry in G of the form a0 + a1ω + . . .+ am−1ωm−1 is

replaced with a0 + a1D + . . .+ am−1Dm−1 to form G (D)

Convolutional code C with polynomial generator matrix G (D)

dLfree(C) ≥ dB(C) ≥ d

Objective 2

C (B, h,G ) construction by Sole and Sison

Let C be a block code over R with polynomial generatormatrix G with minimum distance d

Each entry in G of the form a0 + a1ω + . . .+ am−1ωm−1 is

replaced with a0 + a1D + . . .+ am−1Dm−1 to form G (D)

Convolutional code C with polynomial generator matrix G (D)

dLfree(C) ≥ dB(C) ≥ d

Objective 2

C (B, h,G ) construction by Sole and Sison

Let C be a block code over R with polynomial generatormatrix G with minimum distance d

Each entry in G of the form a0 + a1ω + . . .+ am−1ωm−1 is

replaced with a0 + a1D + . . .+ am−1Dm−1 to form G (D)

Convolutional code C with polynomial generator matrix G (D)

dLfree(C) ≥ dB(C) ≥ d

Objective 2

C (B, h,G ) construction by Sole and Sison

Let C be a block code over R with polynomial generatormatrix G with minimum distance d

Each entry in G of the form a0 + a1ω + . . .+ am−1ωm−1 is

replaced with a0 + a1D + . . .+ am−1Dm−1 to form G (D)

Convolutional code C with polynomial generator matrix G (D)

dLfree(C) ≥ dB(C) ≥ d

Objective 2

C (B, h,G ) construction by Sole and Sison

Let C be a block code over R with polynomial generatormatrix G with minimum distance d

Each entry in G of the form a0 + a1ω + . . .+ am−1ωm−1 is

replaced with a0 + a1D + . . .+ am−1Dm−1 to form G (D)

Convolutional code C with polynomial generator matrix G (D)

dLfree(C) ≥ dB(C) ≥ d