Construction of LDPC (Low Density Parity Check) codes using GRS (Generalized Reed-Solomon) code

1. A method for constructing an LDPC (Low Density Parity Check) code using a GRS (Generalized Reed-Solomon) code, the method comprising:

selecting a location set that comprises:

a plurality of elements, wherein each element of the plurality of elements is generated using a primitive element of a Galois field raised to a corresponding exponent, and wherein the Galois field comprises a predetermined finite number of elements;

a first degree 1 polynomial function that is operable to map each element of the plurality of elements of the location set to a corresponding non-zero value; and

a second degree 1 polynomial function that is a non-linear scalar multiple of the first degree 1 polynomial function;

selecting a plurality of non-zero elements from the Galois field;

identifying a first codeword vector of a GRS code, from among a plurality of possible codeword vector values, wherein the first codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the first degree 1 polynomial function;

identifying a second codeword vector of the GRS code, from among the plurality of possible codeword vector values, wherein the second codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the second degree 1 polynomial function;

multiplying the first codeword vector by each scaling factor of a first plurality of scaling factors thereby generating a plurality of scaled first codeword vectors;

multiplying the second codeword vector by each scaling factor of a second plurality of scaling factors thereby generating a plurality of scaled second codeword vectors;

generating a plurality of cosets by adding each scaled first codeword vector of the plurality of scaled first codeword vectors to each plurality of scaled second codeword vectors of the plurality of scaled second codeword vectors;

generating a plurality of permutation matrices, wherein each permutation matrix of the plurality of permutation matrices comprises a plurality of rows such that each row of the plurality of rows comprises a location mapping of one coset of the plurality of cosets; and

arranging each permutation matrix of the plurality of permutation matrices as sub-matrices thereby generating an LDPC parity check matrix that corresponds to the LDPC code;

and wherein:

the method is performed with an apparatus that provides the generated LDPC parity check matrix to at least one of an encoder and a decoder.

2. The method of claim 1 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the plurality of cosets comprises 2 s cosets; and

each permutation matrix of the plurality of permutation matrices is a 2 s ×2 s permutation matrix.

3. The method of claim 1 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers; and

ρ is less than or equal to 2 s .

4. The method of claim 1 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the LDPC parity check matrix has a bit node degree of γ and a check node degree of ρ;

each column of the LDPC parity check matrix comprises a weight of γ;

each row of the LDPC parity check matrix comprises a weight of ρ; and

ρ is less than or equal to 2 s .

5. The method of claim 1 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 6 elements;

the LDPC parity check matrix is a 384×2048 matrix;

the LDPC code is a (2048,1723) regular LDPC code that comprises a code rate of approximately 0.8413; and

a minimum distance of the LDPC code is greater than or equal to 8.

6. The method of claim 1 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the first plurality of scaling factors comprises 2 s scaling factors; and

the second plurality of scaling factors comprises 2 s scaling factors.

7. The method of claim 1 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the plurality of elements of the location set comprises ρ elements; and

the plurality of non-zero elements comprises ρ elements.

8. The method of claim 1 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

a minimum distance of the LDPC code is greater than or equal to γ+2 when γ is an even integer; and

the minimum distance of the LDPC code is greater than or equal to γ+1 when γ is an odd integer.

9. The method of claim 1 , wherein:

the first plurality of scaling factors is the second plurality of scaling factors.

10. The method of claim 1 , wherein:

the LDPC code is operable to generate LDPC coded signals that are compliant with recommended practices provided by IEEE (Institute of Electrical & Electronics Engineers) P802.3an (10GBASE-T) Task Force.

11. The method of claim 10 , wherein:

the LDPC parity check matrix that corresponds to the LDPC code is a 384×2048 LDPC parity check matrix, Hb, as provided within Appendix B.

12. A method for constructing an LDPC (Low Density Parity Check) code using a GRS (Generalized Reed-Solomon) code, the method comprising:

identifying a first codeword vector of a GRS code, from among a plurality of possible codeword vector values;

identifying a second codeword vector of the GRS code, from among the plurality of possible codeword vector values;

multiplying the first codeword vector by each scaling factor of a first plurality of scaling factors thereby generating a plurality of scaled first codeword vectors;

multiplying the second codeword vector by each scaling factor of a second plurality of scaling factors thereby generating a plurality of scaled second codeword vectors;

generating a plurality of cosets by adding each scaled first codeword vector of the plurality of scaled first codeword vectors to each plurality of scaled second codeword vectors of the plurality of scaled second codeword vectors;

generating a plurality of permutation matrices, wherein each permutation matrix of the plurality of permutation matrices comprises a plurality of rows such that each row of the plurality of rows comprises a location mapping of one coset of the plurality of cosets; and

arranging each permutation matrix of the plurality of permutation matrices as sub-matrices thereby generating an LDPC parity check matrix that corresponds to the LDPC code;

and wherein:

the method is performed within an apparatus that provides the generated LDPC parity check matrix to at least one of an encoder and a decoder.

13. The method of claim 12 , further comprising:

selecting a location set that comprises:

a plurality of elements, wherein each element of the plurality of elements is generated using a primitive element of a Galois field raised to a corresponding exponent, and wherein the Galois field comprises a predetermined finite number of elements;

a first degree 1 polynomial function that is operable to map each element of the plurality of elements of the location set to a corresponding non-zero value; and

a second degree 1 polynomial function that is a non-linear scalar multiple of the first degree 1 polynomial function;

selecting a plurality of non-zero elements from the Galois field; and wherein:

the first codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the first degree 1 polynomial function; and

the second codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the second degree 1 polynomial function.

14. The method of claim 13 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the plurality of cosets comprises 2 s cosets;

each permutation matrix of the plurality of permutation matrices is a 2 s ×2 s permutation matrix;

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the LDPC parity check matrix has a bit node degree of γ and a check node degree of ρ;

each column of the LDPC parity check matrix comprises a weight of γ;

each row of the LDPC parity check matrix comprises a weight of ρ; and

ρ is less than or equal to 2 s .

15. The method of claim 13 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 6 elements;

the LDPC parity check matrix is a 384×2048 matrix;

the LDPC code is a (2048,1723) regular LDPC code that comprises a code rate of approximately 0.8413; and

a minimum distance of the LDPC code is greater than or equal to 8.

16. The method of claim 13 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the first plurality of scaling factors comprises 2 s scaling factors; and

the second plurality of scaling factors comprises 2 s scaling factors.

17. The method of claim 13 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the plurality of elements of the location set comprises ρ elements; and

the plurality of non-zero elements comprises ρ elements.

18. The method of claim 12 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

a minimum distance of the LDPC code is greater than or equal to γ+2 when γ is an even integer;

the minimum distance of the LDPC code is greater than or equal to γ+1 when γ is an odd integer.

19. The method of claim 12 , wherein:

the LDPC code is operable to generate LDPC coded signals that are compliant with recommended practices provided by IEEE (Institute of Electrical & Electronics Engineers) P802.3an (10GBASE-T) Task Force.

20. The method of claim 19 , wherein:

the LDPC parity check matrix that corresponds to the LDPC code is a 384×2048 LDPC parity check matrix, Hb, as provided within Appendix B.

21. An apparatus that is operable to construct an LDPC (Low Density Parity Check) code using a GRS (Generalized Reed-Solomon) code, the apparatus comprising:

a codeword vector identification module that is operable to:

identify a first codeword vector of a GRS code, from among a plurality of possible codeword vector values; and

identify a second codeword vector of the GRS code, from among the plurality of possible codeword vector values;

a coset generation module that is operable to:

multiply the first codeword vector by each scaling factor of a first plurality of scaling factors thereby generating a plurality of scaled first codeword vectors;

multiply the second codeword vector by each scaling factor of a second plurality of scaling factors thereby generating a plurality of scaled second codeword vectors; and

generate a plurality of cosets by adding each scaled first codeword vector of the plurality of scaled first codeword vectors to each plurality of scaled second codeword vectors of the plurality of scaled second codeword vectors;

an LDPC parity check matrix generation module that is operable to:

generate a plurality of permutation matrices, wherein each permutation matrix of the plurality of permutation matrices comprises a plurality of rows such that each row of the plurality of rows comprises a location mapping of one coset of the plurality of cosets; and

arrange each permutation matrix of the plurality of permutation matrices as sub-matrices thereby generating an LDPC parity check matrix that corresponds to the LDPC code.

22. The apparatus of claim 21 , further comprising:

a location set selection module that is operable to select a location set that comprises:

a plurality of elements, wherein each element of the plurality of elements is generated using a primitive element of a Galois field raised to a corresponding exponent, and wherein the Galois field comprises a predetermined finite number of elements;

a first degree 1 polynomial function that is operable to map each element of the plurality of elements of the location set to a corresponding non-zero value; and

a second degree 1 polynomial function that is a non-linear scalar multiple of the first degree 1 polynomial function; and

a non-zero element selection module that is operable to select a plurality of non-zero elements from the Galois field; and wherein:

the first codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the first degree 1 polynomial function; and

the second codeword vector is generated using the plurality of non-zero elements and resultants generated by mapping each element of the plurality of elements of the location set according to the second degree 1 polynomial function.

23. The apparatus of claim 22 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the plurality of cosets comprises 2 s cosets;

each permutation matrix of the plurality of permutation matrices is a 2 s ×2 s permutation matrix;

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the LDPC parity check matrix has a bit node degree of γ and a check node degree of ρ;

each column of the LDPC parity check matrix comprises a weight of γ;

each row of the LDPC parity check matrix comprises a weight of ρ; and

ρ is less than or equal to 2 s .

24. The apparatus of claim 22 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 6 elements;

the LDPC parity check matrix is a 384×2048 matrix;

the LDPC code is a (2048,1723) regular LDPC code that comprises a code rate of approximately 0.8413; and

a minimum distance of the LDPC code is greater than or equal to 8.

25. The apparatus of claim 22 , wherein:

the predetermined finite number of elements of the Galois field comprises 2 s elements such that s is an integer;

the first plurality of scaling factors comprises 2 s scaling factors; and

the second plurality of scaling factors comprises 2 s scaling factors.

26. The apparatus of claim 22 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

the plurality of elements of the location set comprises ρ elements; and

the plurality of non-zero elements comprises ρ elements.

27. The apparatus of claim 21 , wherein:

the LDPC parity check matrix comprises γ×ρ permutation matrices arranged as sub-matrices such that γ and ρ are integers;

a minimum distance of the LDPC code is greater than or equal to γ+2 when γ is an even integer;

the minimum distance of the LDPC code is greater than or equal to γ+1 when γ is an odd integer.

28. The apparatus of claim 21 , wherein:

the apparatus is communicatively coupled to an LDPC encoder;

the LDPC encoder is operable to encode at least one information bit, using the LDPC code constructed by the apparatus, thereby generating an LDPC coded signal that is compliant with recommended practices provided by IEEE (Institute of Electrical & Electronics Engineers) P802.3an (10GBASE-T) Task Force.

29. The apparatus of claim 28 , wherein:

the LDPC parity check matrix that corresponds to the LDPC code is a 384×2048 LDPC parity check matrix, Hb, as provided within Appendix B.