Methods for coding and decoding LDPC codes, and method for forming LDPC parity check matrix

A method for coding LDPC codes, comprising the steps of:

• preparing a parity generation matrix;
• and
• forming a code vector by multiplying an information vector, which is data desired to be coded, by the parity generation matrix, wherein the parity generation matrix meets the following Equation:
• O=aGHT,
• where, O is a zero matrix, a is an arbitrary information vector, G is the parity generation matrix, and H is a parity check matrix, and
• wherein the parity check matrix has a degree distribution G(x) that meets the following Equation:
• G
• ⁡
• (
• x
• )
• =
• ∑
• d
• max
• k
• =
• 2
• ⁢
• C
• ⁡
• (
• k
• +
• α
• )
• -
• γ
• ⁢
• x
• k
• where, C is a parameter that is determined by a normalization condition, G(1)=1, and aα, γ are parameters that are optimized through numerical calculations.

A method for decoding LDPC codes, comprising the steps of:

• preparing a parity check matrix;
• and
• obtaining an information vector from the parity check matrix and a received code vector,
• wherein the parity check matrix has a degree distribution G(x) that meets the following Equation:
• G
• ⁡
• (
• x
• )
• =
• ∑
• d
• i
• -
• 1
• k
• =
• 2
• ⁢
• a
• k
• ⁢
• x
• k
• +
• ∑
• d
• max
• k
• =
• d
• i
• ⁢
• Ck
• -
• γ
• ⁢
• x
• k
• where, ak is a parameter that corresponds to the probability that nodes of the graph have a degree k, C is a parameter that is determined by a normalization condition, G(1)=1 and γ is a parameter that is optimized through numerical calculations.

A method for decoding LDPC codes, comprising the steps of:

• preparing a parity check matrix;
• and
• obtaining an information vector from the parity check matrix and a received code vector,
• wherein the parity check matrix has a degree distribution G(x) that meets the following Equation:
• G
• ⁡
• (
• x
• )
• =
• ∑
• d
• max
• k
• =
• 2
• ⁢
• C
• ⁡
• (
• k
• +
• α
• )
• -
• γ
• ⁢
• x
• k
• where, C is a parameter that is determined by a normalization condition, G(1)=1, and γ is a parameter that is optimized through numerical calculations.

A method for forming an LDPC parity check matrix, comprising the steps of:

• preparing a plurality of parity check matrixes;
• and
• selecting a parity check matrix having maximum performance of the prepared parity check matrixes,
• wherein the parity check matrix has a degree distribution G(x) that meets the following Equation:
• G
• ⁡
• (
• x
• )
• =
• ∑
• d
• i
• -
• 1
• k
• =
• 2
• ⁢
• a
• k
• ⁢
• x
• k
• +
• ∑
• d
• max
• k
• =
• d
• i
• ⁢
• Ck
• -
• γ
• ⁢
• x
• k
• where, ak is a parameter that corresponds to the probability that nodes of the graph have a degree k, C is a parameter that is determined by a normalization condition, G(1)=1, and α, γ are parameters that is optimized through numerical calculations.

A method for forming an LDPC parity check matrix, comprising the steps of:

• preparing a plurality of parity check matrixes;
• and
• selecting a parity check matrix having maximum performance of the prepared parity check matrixes,
• wherein the parity check matrix has a degree distribution G(x) that meets the following Equation:
• G
• ⁡
• (
• x
• )
• =
• ∑
• d
• max
• k
• =
• 2
• ⁢
• C
• ⁡
• (
• k
• +
• α
• )
• -
• γ
• ⁢
• x
• k
• where, C is a parameter that is determined by a normalization condition, G(1)=1, and α, γ are parameters that is optimized through numerical calculations.