Coding method and decoding method
An encoding method generates an encoded sequence by performing encoding of a given coding rate according to a predetermined parity check matrix. The predetermined parity check matrix is a first parity check matrix or a second parity check matrix. The first parity check matrix corresponds to a low-density parity check (LDPC) convolutional code using a plurality of parity check polynomials. The second parity check matrix is generated by performing at least one of row permutation and column permutation with respect to the first parity check matrix. An eth parity check polynomial that satisfies zero, of the LDPC convolutional code, is expressible by using a predetermined mathematical formula.
1. A recording method for an optical disc, the recording method comprising:
generating an encoded sequence using encoding circuitry of a recording device, the encoded sequence comprising: n−1 information sequences denoted as X 1 through X n−1 ; and a parity sequence denoted as P, by encoding the n−1 information sequences at a (n−1)/n coding rate according to a predetermined parity check matrix having m×z rows and n×m×z columns, n being an integer no less than two, m being an even number no less than two, and z being a natural number;
generating a recording pattern by modulating the encoded sequence using modulation circuitry of the recording device; and
recording the recording pattern to an optical disc using an optical pick-up of the recording device, wherein
the encoding circuitry includes a shift register in which the n−1 information sequences are inputted,
in the generating the encoded sequence, the parity sequence is calculated using values held in the shift register and the predetermined parity check matrix,
the predetermined parity check matrix is a first parity check matrix or a second parity check matrix, the first parity check matrix corresponding to a low-density parity check (LDPC) convolutional code using a plurality of parity check polynomials, the second parity check matrix generated by performing at least one of row permutation and column permutation with respect to the first parity check matrix, and
given e denoting an integer no less than zero and no greater than m×z−1, α denoting an integer no less than one and no greater than m×z, and i being a variable denoting an integer that is no less than zero and no greater than m−1 and satisfies i=e % m where % denotes a modulo operator,
when e≠α−1, an eth parity check polynomial that satisfies zero, of the LDPC convolutional code, is expressed as
(
D
b
1
,
i
+
1
)
P
(
D
)
+
∑
k
=
1
n
-
1
{
(
1
+
∑
j
=
1
r
k
,
i
D
a
k
,
i
,
j
)
X
k
(
D
)
}
=
0
(
Math
.
1
)
where b 1,i is a natural number, and
when e=α−1, the eth parity check polynomial that satisfies zero, of the LDPC convolutional code, is expressed as
P
(
D
)
+
∑
k
=
1
n
-
1
{
(
1
+
∑
j
=
1
r
k
,
i
D
a
k
,
i
,
j
)
X
k
(
D
)
}
=
0
(
Math
.
2
)
where, in Math. 1 and Math. 2,
k denotes an integer no less than one and no greater than n−1, j denotes an integer no less than one and no greater than r k,i , and r k,i denotes an integer no less than two,
D denotes a delay operator, X k (D) denotes a polynomial representation of an information sequence Xk among the n−1 information sequences, and P(D) denotes a polynomial representation of the parity sequence P, and
a k,i,j denotes a natural number, and
when x and y are integers no less than one and no greater than r k,i and satisfy x≠y, a k,i,x ≠a k,i,y holds true for all x and y, and
when v k,1 and v k,2 are odd numbers less than m, a k,i,j satisfies both a k,i,1 % m=v k,1 and a k,i,2 % m=v k,2 for all i.
2. A playback method for an optical disc, the playback method comprising:
reading out a recording pattern from an optical disc using an optical pick-up of a playback device;
generating an encoded sequence by demodulating the recording pattern using demodulating circuitry of the playback device.
decoding the encoded sequence according to a predetermined parity check matrix and by employing belief propagation (BP) using error correction decoding circuitry of the playback device, wherein
the encoded sequence comprises: n−1 information sequences denoted as X 1 through X n−1 ;
and a parity sequence denoted as P, the encoded sequence having been generated by encoding the n−1 information sequences at a (n−1)/n coding rate according to the predetermined parity check matrix, the predetermined parity check matrix having m×z rows and n×m×z columns, n being an integer no less than two, m being an even number no less than two, and z being a natural number,
an encoding device, which generates the recording pattern, includes a shift register in which the n−1 information sequences are inputted, and the parity sequence is calculated using values held in the shift register and the predetermined parity check matrix,
the predetermined parity check matrix is a first parity check matrix or a second parity check matrix, the first parity check matrix corresponding to a low-density parity check (LDPC) convolutional code using a plurality of parity check polynomials, the second parity check matrix generated by performing at least one of row permutation and column permutation with respect to the first parity check matrix, and
given e denoting an integer no less than zero and no greater than m×z−1, α denoting an integer no less than one and no greater than m×z, and i being a variable denoting an integer that is no less than zero and no greater than m−1 and satisfies i=e % m where % denotes a modulo operator,
when e≠α−1, an eth parity check polynomial that satisfies zero, of the LDPC convolutional code, is expressed as
(
D
b
1
,
i
+
1
)
P
(
D
)
+
∑
k
=
1
n
-
1
{
(
1
+
∑
j
=
1
r
k
,
i
D
a
k
,
i
,
j
)
X
k
(
D
)
}
=
0
(
Math
.
1
)
where b 1,i is a natural number, and
when e=α−1, the eth parity check polynomial that satisfies zero, of the LDPC convolutional code, is expressed as
P
(
D
)
+
∑
k
=
1
n
-
1
{
(
1
+
∑
j
=
1
r
k
,
i
D
a
k
,
j
,
i
)
X
k
(
D
)
}
=
0
(
Math
.
2
)
where, in Math. 1 and Math. 2,
k denotes an integer no less than one and no greater than n−1, j denotes an integer no less than one and no greater than r k,i , and r k,i denotes an integer no less than two,
D denotes a delay operator, X k (D) denotes a polynomial representation of an information sequence X k among the n−1 information sequences, and P(D) denotes a polynomial representation of the parity sequence P, and
a k,i,j denotes a natural number, and
when x and y are integers no less than one and no greater than r k,i and satisfy x≠y, a k,i,x ≠a k,i,y holds true for all x and y, and
when v k,1 and v k,2 are odd numbers less than m, a k,i,j satisfies both a k,i,1 % m=v k,1 and a k,i,2 % m=v k,2 for all i.