LDPC design for high parallelism, low error floor, and simple encoding
A method of data encoding is disclosed. An encoder receives a set of information bits and performs an LDPC encoding operation on the set of information bits to produce a codeword based on a matched lifted LDPC code. The matched lifted LDPC code is based on a commutative lifting group and includes a number of parity bits and a submatrix to determine values of the parity bits. An order of the lifting group (Z) corresponds with a size of the lifting. A determinant of the submatrix is a polynomial of the form: g a +(g 0 +g L )P, where g 0 is the identity element of the group, g 0 =g L 2 k , and P is an arbitrary non-zero element of a binary group ring associated to the lifting group.
1. A method of data encoding, the method comprising:
receiving a set of information bits;
performing a low density parity check (LDPC) encoding operation on the set of information bits to produce a codeword based on a matched lifted LDPC code, wherein the matched lifting is based on a commutative lifting group and wherein the LDPC code includes a number of parity bits and a submatrix to determine values of the parity bits, and wherein:
an order of the lifting group (Z) corresponds with a size of the lifting;
a determinant of the submatrix is a polynomial of the form: g a +(g 0 +g L )P, wherein:
g 0 is the identity element of the group,
g 0 =g L 2 k ,
Z=2 k *L, for a positive integers k and L,
a=2 l *L, for a non-negative l<k, and
P is an arbitrary non-zero element of a binary group ring associated to the lifting group.
2. The method of claim 1 wherein the lifting group is a cyclic group, wherein:
g i may be identified with x i ; and
the determinant of the submatrix is of the form: x a +(1+x L )P(x), wherein P(x) has at least two terms and 2 k L=0 modulo Z.
3. The method of claim 1 in which the submatrix is, up to permutations of its rows and columns, upper triangular.
4. The method of claim 3 , wherein elements below the first subdiagonal of the submatrix are equal to 0.
5. A non-transitory computer-readable storage medium containing program instructions that, when executed by a processor provided within a communications device, causes the device to:
receive a set of information bits;
perform an LDPC encoding operation on the set of information bits to produce a codeword based on a matched lifted LDPC code, wherein the matched lifting is based on a commutative lifting group and wherein the LDPC code includes a number of parity bits and a submatrix to determine values of the parity bits, and wherein:
an order of the lifting group (Z) corresponds with a size of the lifting;
a determinant of the submatrix is a polynomial of the form: g a +(g 0 +g L )P, wherein:
g 0 is the identity element of the group,
g 0 =g L 2 k ,
z=2 k *L, for a positive integers k and L,
a=2 l *L, for a non-negative l<k, and
P is an arbitrary non-zero element of a binary group ring associated to the lifting group.
6. The non-transitory computer-readable medium of claim 5 wherein the lifting group is a cyclic group, wherein:
g i may be identified with x i ; and
the determinant of the submatrix is of the form: x a +(1+x L )P(x), wherein P(x) has at least two terms and 2 k L=0 modulo Z.
7. The non-transitory computer-readable medium of claim 5 in which the submatrix is, up to permutations of its rows and columns, upper triangular.
8. The non-transitory computer-readable medium of claim 7 , wherein elements below the first subdiagonal of the submatrix are equal to 0.
9. A communications device, comprising:
a memory to store a set of information bits; and
an encoder to:
perform an LDPC encoding operation on the set of information bits to produce a codeword based on a matched lifted LDPC code, wherein the matched lifting is based on a commutative lifting group and wherein the LDPC code includes a number of parity bits and a submatrix to determine values of the parity bits, and wherein:
an order of the lifting group (Z) corresponds with a size of the lifting;
a determinant of the submatrix is a polynomial of the form: g a +(g 0 +g L )P, wherein:
g 0 is the identity element of the group,
g 0 =g L 2 k ,
z=2 k *L, for a positive integers k and L,
a=2 l *L, for a non-negative l<k, and
P is an arbitrary non-zero element of a binary group ring associated to the lifting group.
10. The device of claim 9 wherein the lifting group is a cyclic group, wherein:
g i may be identified with x i ; and
the determinant of the submatrix is of the form: x a +(1+X L )P(x), wherein P(x) has at least two terms and 2 k L=0 modulo Z.
11. The device of claim 9 in which the submatrix is, up to permutations of its rows and columns, upper triangular.
12. The device of claim 11 , wherein elements below the first subdiagonal of the submatrix are equal to 0.
13. An encoder, comprising:
means for receiving a set of information bits;
means for performing an LDPC encoding operation on the set of information bits to produce a codeword based on a matched lifted LDPC code, wherein the matched lifting is based on a commutative lifting group and wherein the LDPC code includes a number of parity bits and a submatrix to determine values of the parity bits, and wherein:
an order of the lifting group (Z) corresponds with a size of the lifting;
a determinant of the submatrix is a polynomial of the form: g a +(g 0 +g L )P, wherein:
g 0 is the identity element of the group,
g 0 =g L 2 k ,
Z=2 k *L, for a positive integers k and L,
a=2 l *L, for a non-negative l<k, and
P is an arbitrary non-zero element of a binary group ring associated to the lifting group.