1. Feng Lu Chuan Heng Foh, Jianfei Cai and Liang- Tien Chia Information Theory, 2009. ISIT 2009. IEEE International Symposium on LT Codes Decoding: Design and Analysis

2. Outline Introduction Full rank LT decoding process LT decoding drawbacks Full rank decoding Recovering the borrowed symbol Non-square case Random matrix rank Random matrix rank when n=k Random matrix rank when n > k Numerical results and discussion

3. Introduction LT codes Large value of k : Perform very well [5] Small numbers of k : Often encountered difficulties [7] optimize the configuration parameters of the degree distribution Only handle symbols k≤10 [9] using Gaussian elimination method for decoding The decoding complexity increase significantly [5] A. Shokrollahi, "Raptor Codes," IEEE Transactions on Information Theory, Vol. 52, no. 6, pp. 2551-2567, 2006. [7] E. Hyytia,T. Tirronen, J. Virtamo, "Optimal Degree Distribution for LT Codes with Small Message Length," The 26th IEEE International Conference on Computer Communications INFOCOM, pp. 2576-2580, 2007. [9] J. Gentle, "Numerical Linear Algebra for Application in Statistics," pp. 87-91, Springer-Verlag, 1998

4. Introduction We propose a new decoding process called full rank decoding algorithm To preserve the low complexity benefit of LT codes : Retaining the original LT encoding and decoding process in maximal possible extent To prevent LT decoding from terminating prematurely: Our proposed method extends the decodability of LT decoding process

5. Full rank LT decoding process LT decoding drawbacks Full rank decoding Recovering the borrowed symbol Non-square case

6. LT decoding drawbacks The LT decoding process terminates when there is no more symbol left in the ripple. When LT decoding process terminates By using Gaussian elimination, often the undecodable packets can be decoded to recover all symbols.

7. LT decoding drawbacks Viewing a packet as an equation formed by combining linearly a number of variables (or symbols) in GF(2) The set of available equations (or packets) may give a full rank A numerical solver (or decoder) can determine all variables (or symbols). Attributing to the design of the LT decoding process, the method recovers only partial but not all symbols

8. GF(2) GF(2) is the Galois field of two elements. The two elements are nearly always 0 and 1. Addition operation : Multiplication operation : +01 00 1 110 ×01 000 101

9. Full rank decoding 1. Whenever the ripple is empty An early termination 2. A particular symbol is borrowed Decoded through some other method 3. Placing the symbol into the ripple for the LT decoding process to continue. 4. Repeated until the LT decoding process terminates with a success In the case of full rank, any picked borrowed symbol can be decoded with a suitable method

10. Full rank decoding Mainly uses LT decoding to recover symbols When LT decoding fails Trigger Wiedemann algorithm to recover a borrowed symbol Return back to LT decoding to recover subsequent symbols How to choose the borrowed symbol ?  Choose the symbol that is carried by most packets

11. Full rank decoding

12. Recovering the borrowed symbol We need to seek for a suitable method that can recover only a single symbol using a low computational cost. Let M denote the coefficient matrix. (n*k) M is defined over GF(2), x: size k*l, y: size n*l

13. Recovering the borrowed symbol We let n=k  We want to solve for a particular symbol. x’: size k*1, describes the selection of row vectors x’: size k*1, where the unique 1 locates at the index i The inner product of (x', y) gives the borrowed symbol.

14. Recovering the borrowed symbol We use the efficient Wiedemann algorithm [11] to solve The vector u, is used to generate Krylov sequence : Let S be the space spanned by this sequence M : the operator M restricted to S : the minimal polynomial of M; (Using the BM algorithm [12], [13]) [I I] D. Wiedemann, "Solving sparse linear equations over finite fields," IEEE Transactions on Information Theory, Vol. 32, no. I, pp. 54-62, 1986. [12] E. Berlekamp, "Algebraic Coding Theory," McGraw-Hili, New York,1968. [13] J. Massey, "Shift-register synthesis and BCII decoding," IEEE Transactions on Information Theory, Vol. 15, no. I, pp. 122-127, 1969.

15. Non-square case n > k The coefficient matrix M will be non-square  Find a n x k matrix Me,such that MjM, will be of full rank M should be of full rank  One way to obtain Me is to randomly set an entry of row i in Me Once x' is solved, the recovered symbol is obtained as

16. Random matrix rank The probability of successful decoding for our proposed algorithm The probability that the coefficient matrix M is of full rank M is of full rank Our proposed algorithm guarantees the success of the decoding.

17. Random matrix rank when n=k Let Vi be the row vector of M. The row vectors are linearly dependent if there exists a nonzero vector (C1,"" Ck) E GF (2 that satisfies If M is said to have a full rank, any linear combination of coefficient vectors (VI, V2,...,Vk) will not produce 0. Consider a non-zero vector c with exactly q non-zero coordinates. Define to be the probability that

18. Random matrix rank when n=k Suppose that summing the first q vectors resulting a vector with degree i. The probability that of degree (a + b) is

19. The state transition probability : This allows us to determine the degree distribution of the sum of any number of vectors. Random matrix rank when n=k

20. We shall define a transition matrix Tr with dimension (k+1) x (k+1) Let denotes the degree distribution of the sum of q vectors (q ≥1)

21. If M is said to have a full rank, any linear combination of coefficient vectors (VI, V2,...,Vk) will not produce 0. : the probability that The probability of full rank Random matrix rank when n=k

22. Random matrix rank when n > k For a full rank matrix, no linear dependency exists for any combination of the row vectors  Which is not true for the case of n > k Let (q, r) denote M consists of q row vectors with rank r

23. Random matrix rank when n > k We can be utilize the methods like eigen decomposition or companion matrix and Jordan normal form [15] to derive a closed form expression for P(q, r). Initialized to [15] R.A. Hom, C.R. Johnson, "Matrix Analysis," Cambridge University Press, 1985

24. Random matrix rank when n > k

25. Numerical results and discussion [6] R. Karp, M. Luby, A. Shokrollahi, “Finite length analysis of LT codes,” The IEEE International Symposium on Information Theory, 2004.