Method of codifying data including generation of a quasi-cyclic code
A method including selecting a factor based on a number of bits in a codeword and a natural number and generating a model matrix including first and second matrices having data and parity bits. Hamming weights of the model matrix are not constant and Hamming weights of columns of the model matrix follow a statistical distribution dependent upon a codification rate of a channel. A compact matrix is generated by replacing elements of the model matrix equal to: 1 with a pseudo-random positive whole number; and 0 with −1. A quasi-cyclic code is generated by replacing in the compact matrix: positive elements with identity matrices; and elements equal to −1 with null matrices. A number of rows and columns in each of the identity and null matrices is equal to the factor. The quasi-cyclic code is applied to a word to generate a codeword, which is transmitted on the channel.
1. A method comprising:
selecting a factor at a first device, wherein the factor is equal to a number of bits in a codeword divided by a natural number;
generating a model matrix, wherein the model matrix includes (i) a first matrix comprising data, and (ii) a second matrix comprising parity bits, wherein Hamming weights of columns and rows of the model matrix are not constant and the Hamming weights of the columns correspond to a codification rate of a channel such that the model matrix is irregular;
generating a compact matrix by replacing (i) each element of the model matrix that is equal to 1 with a pseudo-random positive whole number, and (ii) each element of the model matrix that is equal to 0 with a −1, wherein the pseudo-random positive whole number is greater than or equal to 0 and less than the factor;
generating a quasi-cyclic code including a parity matrix by replacing (i) each positive element of the compact matrix with an identity matrix, and (ii) each element of the compact matrix equal to −1 with a null matrix, wherein each of the identity matrices is rotated cyclically a number of times indicated by a corresponding one of the positive elements of the compact matrix, wherein a number of rows in each of the identity matrices and each of the null matrices is equal to the factor, and wherein a number of columns in each of the identity matrices and each of the null matrices is equal to the factor;
applying the quasi-cyclic code to a word of the data via a codification device to generate the codeword; and
transmitting the codeword on the channel from the first device to a second device.
2. The method of claim 1 , wherein:
a number of rows in the parity matrix is equal to a number of bits in the quasi-cyclic code minus a number of bits in the word of the data; and
a number of columns in the parity matrix is equal to the number of bits in the quasi-cyclic code.
3. The method of claim 1 , wherein:
a first portion of the second matrix of the model matrix includes a column vector;
a number of elements in the column vector is based on a difference between a first parameter and a second parameter;
the first parameter is equal to a number of bits in the quasi-cyclic code divided by the factor; and
the second parameter is equal to a number of bits in the word of the data divided by the factor.
4. The method of claim 3 , wherein:
a second portion of the second matrix of the model matrix comprises a triple diagonal structure;
the triple diagonal structure includes (i) a first central diagonal, (ii) a second central diagonal, and (iii) a last row diagonal; and
bits in the first central diagonal, the second central diagonal, and the last row diagonal are equal to 1 and each of a remainder of bits in the triple diagonal structure is equal to 0.
5. The method of claim 1 , wherein:
a portion of the second matrix of the model matrix comprises a triple diagonal structure;
the triple diagonal structure includes (i) a first central diagonal, (ii) a second central diagonal, and (iii) a last row diagonal; and
bits in the first central diagonal, the second central diagonal, and the last row diagonal are equal to 1 and each of a remainder of bits in the triple diagonal structure is equal to 0.
6. The method of claim 1 , wherein:
a first portion of the second matrix of the model matrix comprises a pseudo-random column vector having a Hamming weight greater than 2;
a second portion of the second matrix of the model matrix comprises a double diagonal structure;
the double diagonal structure includes (i) a first central diagonal, and (ii) a second central diagonal; and
bits in the first central diagonal and the second central diagonal are equal to 1 and each of a remainder of bits in the double diagonal structure is equal to 0.
7. The method of claim 6 , wherein the pseudo-random positive whole numbers that replace the elements of the model matrix that are equal to 1 are a same pseudo-random positive whole number.
8. The method of claim 1 , wherein the pseudo-random positive whole numbers that replace the elements of the model matrix that are equal to 1 are different pseudo-random positive whole numbers.
9. The method of claim 1 , wherein:
the quasi-cyclic code is a low density parity check code; and
the codification rate is equal to a number of bits in the word of the data divided by a number of bits in the quasi-cyclic code.
10. The method of claim 1 , wherein a transpose of the quasi-cyclic code multiplied by the codeword is equal to 0.
11. The method of claim 1 , wherein:
a transpose of the quasi-cyclic code multiplied by a binary matrix is equal to 0;
a number of rows in the binary matrix is equal to a number of bits in the word of the data; and
a number of columns in the binary matrix is equal to a number of bits in the codeword.
12. The method of claim 1 , wherein the model matrix comprises:
1
0
1
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
1
0
1
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1.
13. The method of claim 1 , wherein the compact matrix comprises:
13
-
1
7
-
1
-
1
1
-
1
1
-
1
-
1
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
5
-
1
-
1
-
1
11
-
1
-
1
-
1
4
-
1
4
6
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
13
-
1
-
1
-
1
-
1
11
-
1
10
-
1
9
13
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
13
-
1
-
1
6
-
1
10
-
1
5
-
1
-
1
4
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
8
8
2
11
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
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1
-
1
-
1
-
1
-
1
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1
-
1
3
-
1
-
1
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1
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1
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1
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1
1
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1
4
1
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1
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1
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1
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1
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1
0
0
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1
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1
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1
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1
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1
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1
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1
4
2
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1
2
9
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
0
0
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1
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1
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1
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1
2
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1
13
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1
4
9
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
0
0
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1
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1
-
1
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1
11
-
1
-
1
-
1
-
1
-
1
-
1
6
4
11
12
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
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1
-
1
-
1
-
1
-
1
10
-
1
-
1
-
1
-
1
1
-
1
13
13
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
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1
0
0
-
1
-
1
-
1
0
-
1
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1
0
2
2
-
1
-
1
-
1
-
1
0
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1
-
1
-
1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
1
2
11
2
4
12
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
14. The method of claim 1 , wherein the compact matrix comprises:
-
1
-
1
-
1
6
-
1
-
1
9
6
-
1
-
1
2
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
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1
0
-
1
-
1
-
1
3
-
1
12
1
-
1
-
1
3
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
9
11
-
1
-
1
13
-
1
-
1
2
12
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
1
-
1
-
1
11
-
1
-
1
7
-
1
-
1
-
1
11
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
4
8
-
1
-
1
-
1
-
1
-
1
2
5
4
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
3
0
-
1
-
1
8
-
1
-
1
1
-
1
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1
-
1
-
1
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1
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1
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1
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1
0
0
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1
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1
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1
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1
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1
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1
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1
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1
0
6
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1
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1
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1
-
1
5
13
-
1
-
1
-
1
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1
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1
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1
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1
0
0
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1
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1
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1
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1
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1
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1
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1
9
1
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1
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1
3
-
1
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1
3
1
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1
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1
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1
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1
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1
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1
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1
0
0
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1
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1
-
1
9
0
13
-
1
-
1
12
-
1
-
1
8
-
1
-
1
-
1
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1
-
1
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1
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1
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1
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1
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1
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1
0
0
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1
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1
-
1
5
-
1
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1
1
4
-
1
-
1
5
-
1
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1
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1
-
1
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1
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1
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1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
8
-
1
-
1
8
-
1
-
1
9
0
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
10
11
-
1
-
1
-
1
3
-
1
-
1
0
-
1
-
1
-
1
4
8
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
15. The method of claim 1 , wherein the compact matrix comprises:
-
1
52
-
1
64
-
1
-
1
60
-
1
-
1
-
1
-
1
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
10
-
1
-
1
-
1
-
1
79
-
1
-
1
79
-
1
78
51
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
9
-
1
-
1
-
1
-
1
-
1
-
1
75
29
72
8
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
52
16
63
-
1
-
1
65
-
1
-
1
-
1
-
1
-
1
40
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
24
-
1
-
1
47
1
39
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
52
-
1
-
1
-
1
-
1
-
1
-
1
-
1
53
79
48
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
0
-
1
-
1
72
-
1
67
57
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
7
-
1
-
1
-
1
2
50
-
1
-
1
-
1
-
1
15
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
15
-
1
19
-
1
-
1
-
1
-
1
-
1
75
51
43
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
72
-
1
-
1
-
1
38
-
1
-
1
-
1
69
-
1
62
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
19
-
1
41
-
1
-
1
1
41
-
1
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
41
-
1
17
-
1
-
1
-
1
-
1
-
1
15
-
1
30
-
1
40
6
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
16. The method of claim 1 , wherein the compact matrix comprises:
27
-
1
-
1
-
1
55
19
-
1
30
-
1
-
1
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
-
1
-
1
-
1
70
-
1
47
-
1
62
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
41
-
1
-
1
-
1
44
-
1
-
1
59
60
25
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
16
77
-
1
-
1
-
1
5
-
1
48
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
45
-
1
27
-
1
46
19
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
63
-
1
-
1
-
1
55
-
1
-
1
-
1
48
26
10
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
42
-
1
21
-
1
58
-
1
41
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
78
0
-
1
7
52
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
29
9
-
1
-
1
-
1
37
-
1
-
1
-
1
35
21
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
22
72
-
1
-
1
47
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
35
-
1
-
1
-
1
-
1
13
-
1
35
-
1
70
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
46
28
-
1
-
1
-
1
38
-
1
-
1
-
1
8
-
1
10
58
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
17. The method of claim 1 , wherein the compact matrix comprises:
-
1
-
1
-
1
-
1
-
1
-
1
297
106
328
-
1
-
1
99
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
290
0
312
-
1
32
-
1
120
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
183
57
-
1
-
1
187
68
-
1
-
1
-
1
-
1
260
-
1
81
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
323
-
1
-
1
-
1
137
354
-
1
-
1
162
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
228
-
1
-
1
-
1
-
1
224
-
1
114
-
1
245
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
113
98
-
1
-
1
120
23
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
138
-
1
187
45
62
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
142
-
1
-
1
-
1
347
67
-
1
-
1
-
1
46
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
328
265
-
1
66
156
96
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
212
184
-
1
-
1
102
-
1
-
1
-
1
-
1
120
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
80
15
-
1
329
153
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
207
70
-
1
7
235
-
1
-
1
-
1
-
1
-
1
-
1
-
1
81
185
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
18. The method of claim 1 , wherein the compact matrix comprises:
-
1
34
-
1
95
-
1
279
-
1
-
1
-
1
-
1
248
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
-
1
0
-
1
-
1
-
1
-
1
134
356
275
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
51
-
1
27
-
1
-
1
-
1
-
1
-
1
22
152
-
1
57
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
124
-
1
290
-
1
281
15
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
340
-
1
99
336
-
1
-
1
1
-
1
-
1
-
1
-
1
33
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
163
-
1
46
-
1
-
1
-
1
-
1
-
1
-
1
306
-
1
86
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
185
-
1
24
-
1
-
1
-
1
94
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
223
-
1
225
325
-
1
-
1
-
1
-
1
-
1
297
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
46
-
1
314
-
1
-
1
-
1
59
-
1
-
1
67
-
1
120
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
121
-
1
-
1
-
1
-
1
161
-
1
303
-
1
264
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
303
-
1
8
-
1
185
-
1
-
1
138
-
1
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
312
-
1
-
1
-
1
100
-
1
-
1
144
-
1
307
33
166
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
-
1
0.
19. The method of claim 1 , wherein the compact matrix comprises:
49
-
1
-
1
21
31
-
1
57
-
1
-
1
19
-
1
29
2
-
1
19
-
1
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
7
22
-
1
-
1
37
-
1
32
10
-
1
26
-
1
-
1
59
-
1
48
-
1
0
0
-
1
-
1
-
1
-
1
-
1
53
-
1
-
1
20
50
-
1
-
1
3
16
-
1
49
-
1
-
1
28
14
-
1
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
58
23
-
1
-
1
15
54
-
1
-
1
5
-
1
18
49
-
1
-
1
13
-
1
-
1
-
1
0
0
-
1
-
1
-
1
55
-
1
-
1
58
-
1
9
-
1
26
57
-
1
41
-
1
31
-
1
21
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
10
49
-
1
59
-
1
7
-
1
-
1
30
-
1
18
-
1
48
-
1
7
59
-
1
-
1
-
1
-
1
0
0
-
1
48
-
1
-
1
50
18
-
1
-
1
11
52
-
1
59
-
1
-
1
37
-
1
10
0
-
1
-
1
-
1
-
1
-
1
0
0
-
1
24
16
-
1
-
1
0
53
-
1
-
1
41
-
1
38
51
-
1
58
-
1
59
8
-
1
-
1
-
1
-
1
-
1
0.
20. The method of claim 1 , wherein the compact matrix comprises:
78
-
1
-
1
167
237
-
1
3
-
1
266
-
1
-
1
102
153
-
1
-
1
212
-
1
0
-
1
-
1
-
1
-
1
-
1
-
1
-
1
83
189
-
1
-
1
68
-
1
178
-
1
90
205
-
1
-
1
13
4
-
1
-
1
0
0
-
1
-
1
-
1
-
1
-
1
-
1
226
147
-
1
46
-
1
-
1
76
-
1
116
-
1
211
-
1
112
-
1
118
-
1
-
1
0
0
-
1
-
1
-
1
-
1
92
-
1
-
1
214
-
1
236
241
-
1
157
-
1
143
-
1
214
-
1
207
-
1
-
1
-
1
-
1
0
0
-
1
-
1
-
1
144
-
1
-
1
258
264
-
1
53
-
1
114
-
1
172
-
1
-
1
82
262
-
1
62
-
1
-
1
-
1
0
0
-
1
-
1
-
1
153
120
-
1
-
1
199
-
1
126
-
1
61
-
1
183
15
-
1
-
1
134
-
1
-
1
-
1
-
1
-
1
0
0
-
1
-
1
100
-
1
141
-
1
36
-
1
17
-
1
156
-
1
124
162
-
1
-
1
57
0
-
1
-
1
-
1
-
1
-
1
0
0
196
-
1
187
-
1
73
-
1
80
-
1
139
-
1
57
-
1
-
1
236
267
-
1
62
-
1
-
1
-
1
-
1
-
1
-
1
0.
21. The method of claim 1 , wherein the compact matrix comprises:
-
1
13
32
47
41
24
-
1
25
22
40
1
31
8
15
20
15
42
30
13
3
-
1
0
-
1
-
1
25
46
15
43
45
29
39
47
23
38
39
12
-
1
21
-
1
38
33
0
0
-
1
39
0
0
-
1
35
45
45
38
14
16
6
11
-
1
18
7
41
35
17
32
45
41
-
1
18
17
0
-
1
0
0
9
32
6
22
26
31
9
8
22
32
40
4
18
40
36
-
1
-
1
23
31
41
39
20
-
1
0.
22. The method of claim 1 , wherein the compact matrix comprises:
-
1
47
146
203
184
112
-
1
116
103
181
3
140
38
68
91
70
191
138
62
14
-
1
0
-
1
-
1
117
203
67
194
206
133
174
212
104
171
176
56
-
1
96
-
1
167
149
4
1
-
1
177
0
0
-
1
153
206
198
173
55
72
28
53
-
1
82
34
186
161
80
144
204
187
-
1
84
77
0
-
1
0
0
44
147
27
83
118
130
41
38
100
146
183
19
85
180
163
-
1
-
1
106
140
185
177
94
-
1
0.
23. The method of claim 1 , further comprising puncturing the codeword to remove bits of the codeword prior to the codeword being transmitted from the first device.
24. The method of claim 23 , wherein the puncturing includes the following puncturing pattern:
pp
1152
(
16
/
18
)
=
[
1
1
…
1
︸
720
0
0
…
0
︸
36
1
1
…
1
︸
360
0
0
…
0
︸
36
]
.
25. The method of claim 23 , wherein the puncturing includes the following puncturing pattern:
pp
5184
(
16
/
18
)
=
[
1
1
…
1
︸
3240
0
0
…
0
︸
162
1
1
…
1
︸
972
0
0
…
0
︸
162
1
1
…
1
︸
648
]
.
26. The method of claim 23 , wherein the puncturing includes the following puncturing pattern:
pp
1152
(
20
/
21
)
=
[
1
1
…
1
︸
720
0
0
…
0
︸
48
1
1
…
1
︸
240
0
0
…
0
︸
96
1
1
…
1
︸
48
]
.
27. The method of claim 23 , wherein the puncturing includes the following puncturing pattern:
pp
5184
(
20
/
21
)
=
[
0
0
…
0
︸
216
1
1
…
1
︸
4320
0
0
…
0
︸
432
1
1
…
1
︸
216
]
.