AM aximum-Likelihood Decoding Algorithm of LT Code with a Small

							A Maximum-Likelihood Decoding
Algorithm of LT Code with a Small
     Fraction of Dense Rows



        Ki-Moon Lee     Hayder Radha
        Department of   Department of Electrical and
        Mathematics     Computer Engineering


           Michigan State University, USA
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                                  Contents




 1. Decoding Models: A (0,1)-Linear System
 2. The Separated Maximum-Likelihood Decoding Algorithms (S-MLDA)
 3. Computational Complexities
 4. RSD Design and Rank Properties
 5. Simulation Results
 6. Conclusion




                 A Maximum-Likelihood Decoding Algorithm of LT Code
                        with a Small Fraction of Dense Rows
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       1. Decoding System: A (0,1)-Liner Systems




 LT Codes                     LDPC Codes




            A Maximum-Likelihood Decoding Algorithm of LT Code
                   with a Small Fraction of Dense Rows
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                  2. The Separated-MLDA


     Gaussian Elimination (GE)




               A Maximum-Likelihood Decoding Algorithm of LT Code
                      with a Small Fraction of Dense Rows
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                    2. The Separated-MLDA (1)



     The MPA: Lower Triangulation




                 A Maximum-Likelihood Decoding Algorithm of LT Code
                        with a Small Fraction of Dense Rows
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                 2. The Separated-MLDA (2)

     The BM’s MLDA on an explicit PHQT




               A Maximum-Likelihood Decoding Algorithm of LT Code
                      with a Small Fraction of Dense Rows
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                   2. The Separated-MLDA (3)

 The MLDA on HQT




               A Maximum-Likelihood Decoding Algorithm of LT Code
                      with a Small Fraction of Dense Rows
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                        2. The Separated-MLDA (4)


     I - Pre-Decoding (bit-level)




                                            Improvement (in post-decoding)


 II - Post-Decoding (symbol-level)

                                                                             Example




                       A Maximum-Likelihood Decoding Algorithm of LT Code
                              with a Small Fraction of Dense Rows
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            2. The Separated-MLDA (5)


         Advantages




                                                               Example




     (Possible) Draw-Back




          A Maximum-Likelihood Decoding Algorithm of LT Code
                 with a Small Fraction of Dense Rows
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                   3. Computational Complexities




      Pre-Decoding (in number of {sign, bit}-flips)
                                                                   Example




      Post-Decoding (in number of symbol additions)
                                                                   Example




                     A Maximum-Likelihood Decoding Algorithm of LT Code
                            with a Small Fraction of Dense Rows
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          4. Degree Design and Rank Properties (1)

      1. The RSD (by Mackay)




                    A Maximum-Likelihood Decoding Algorithm of LT Code
                           with a Small Fraction of Dense Rows
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        4. Degree Design and Rank Properties (2)

 2. RSD with Small Fraction of Dense Rows




      Example with S+1=15, n=1000




                                             Example




                  A Maximum-Likelihood Decoding Algorithm of LT Code
                         with a Small Fraction of Dense Rows
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           4. Degree Design and Rank Properties (3)

      3. The Dense Fraction by Union Bound




      (FR): Kovalenko’s Rank-Distribution




                    A Maximum-Likelihood Decoding Algorithm of LT Code
                           with a Small Fraction of Dense Rows
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          4. Degree Design and Rank Properties

 (FR): Rank Problem




                 A Maximum-Likelihood Decoding Algorithm of LT Code
                        with a Small Fraction of Dense Rows
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      4. Degree Design and Rank Properties

       (FR): EX |Ker(H)|




             A Maximum-Likelihood Decoding Algorithm of LT Code
                    with a Small Fraction of Dense Rows
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              5. Simulation Results (1)

      1. The MPA vs. the S-MLDA in Decoding Failure Rate




           A Maximum-Likelihood Decoding Algorithm of LT Code
                  with a Small Fraction of Dense Rows
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              5. Simulation Results (2)

      2. Number of Symbol Additions by the Post-Decoding




           A Maximum-Likelihood Decoding Algorithm of LT Code
                  with a Small Fraction of Dense Rows
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          5. Simulation Results (3)

      3. Number of References by the S-MLDA




      A Maximum-Likelihood Decoding Algorithm of LT Code
             with a Small Fraction of Dense Rows
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              5. Simulation Results (4)

      4. Rank-Deficient Cases: Number of Free Variables




          A Maximum-Likelihood Decoding Algorithm of LT Code
                 with a Small Fraction of Dense Rows
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              Simulation Results (5)

      5. Comparison in Rank-Deficiency when n=5000




         A Maximum-Likelihood Decoding Algorithm of LT Code
                with a Small Fraction of Dense Rows
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                              Conclusion



      1. The computational complexity of the S-MLDA is
         very efficient compared to conventional GE.

      2. Even for short block lengths and overheads closed
         to 0, successful decoding of codes is possible
         under the S-MLDA , when the RSD is supplemented
         with a small fraction of dense rows.




               A Maximum-Likelihood Decoding Algorithm of LT Code
                      with a Small Fraction of Dense Rows
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                 Thank You




               Questions?




      A Maximum-Likelihood Decoding Algorithm of LT Code
             with a Small Fraction of Dense Rows