LDPC Codes Based on Cyclotomic Cosets Mina Sartipi and Faramarz Fekri Georgia Institute of Technology School of Electric

1. LDPC Codes Based on Cyclotomic CosetsMina Sartipi and Faramarz FekriGeorgia Institute of TechnologySchool of Electrical & Computer EngineeringOctober 2003

2. Outline • Review of LDPC code • Construction of cyclic LDPC codes using cyclotomic cosets • Extending the original cyclic code by column splitting • Concluding remarks Georgia Institute of Technology Center for Signal and Image Processing

3. LDPC Code LDPC (Low density parity check) code: A linear code with a low dense parity check matrix Tanner graph representation: 111000 001100 H= 000111 Check node Variable node Georgia Institute of Technology Center for Signal and Image Processing Variable node

4. 1 0 1 1 1 0 1 0 0 1 0 1 Cycle Four in Tanner Graph A path of length four that closes back on itself: Cycle four degrades the performance. Georgia Institute of Technology Center for Signal and Image Processing

5. Cyclotomic Cosets The cyclotomic cosets modulo nwith respect to GF(2) are a partitioning of the integers {0,1,…,n} into sets of the form x mod (n), x.2 mod (n), … , x. mod(n) Cyclotomic cosets are sets of powers of , be a primitive element in GF( ),in the conjugacy classes. The parity polynomial must have one or more of these minimal polynomials as factors. Question: Which cyclotomic coset shall we choose? Georgia Institute of Technology Center for Signal and Image Processing

6. Design Objective Design goal: Avoid cycles of length four Lemma: If the difference of every two elements of a cyclotomic coset is distinct, then the parity check matrix H that is generated based on this cyclotomic coset does not contain any cycles of length four. Georgia Institute of Technology Center for Signal and Image Processing

7. Code Construction Cyclic LDPC code of length Cyclotomic coset ={ }. If > 2 , then cycles 4 are avoided by lemma. Therefore, the cyclotomic coset is {1,2 ,4,…, } Georgia Institute of Technology Center for Signal and Image Processing

8. Properties of the Generated Cyclic LDPC Codes Georgia Institute of Technology Center for Signal and Image Processing

9. Simulation Results for Cyclic LDPC Code Georgia Institute of Technology Center for Signal and Image Processing

10. H I I I Improvement on the Minimum Distance Minimum distance of the cyclic code = O( log(n)) Minimum distance Error floor Increase minimum distance Add identity matrices at the bottom of the original parity-check matrix Georgia Institute of Technology Center for Signal and Image Processing

11. Simulation Results for Improved Minimum Distance Georgia Institute of Technology Center for Signal and Image Processing

12. Column Splitting Splitting the non-zero elements of one column into two columns by the following rule: Non-zero elements of each column is split between the two new columns based on the elements of the corresponding rows of the Latin-square matrix. Georgia Institute of Technology Center for Signal and Image Processing

13. Column Splitting Based on Latin-Square Matrix Example: 1 = 3 1 1 0 H = 0 1 1 2 = 1 0 1 1 3 = 0 0 0 1 0 1 = 2 1 1 0 0 0 0 0 0 1 0 1 0 1 2 2 1 L = 2 1 1 2 Georgia Institute of Technology Center for Signal and Image Processing

14. Simulation Results for Extended Cyclic LDPC Code Georgia Institute of Technology Center for Signal and Image Processing

15. Conclusion • Proposing cyclic LDPC codes that • Has Linear encoding complexity, • Has the same or even better performance than random LDPC code at short to moderate lengths, and • Contains no cycle of length four. • Improving minimum distance of the proposed cyclic LDPC code that outperforms the random LDPC code. • Extending the original cyclic code by column splitting that • Performs almost one order of magnitude better than random LDPC codes at BER below, . • Contains no cycle of length four. Georgia Institute of Technology Center for Signal and Image Processing