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LT Codes Decoding: Design and Analysis

LT Codes Decoding: Design and Analysis. Feng Lu Chuan Heng Foh , Jianfei Cai and Liang- Tien Chia Information Theory, 2009. ISIT 2009. IEEE International Symposium on . Outline . Introduction Full rank LT decoding process LT decoding drawbacks Full rank decoding

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LT Codes Decoding: Design and Analysis

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  1. LT Codes Decoding: Design and Analysis Feng Lu Chuan HengFoh, JianfeiCai and Liang- TienChia Information Theory, 2009. ISIT 2009. IEEE International Symposium on 

  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. [5] A. Shokrollahi, "Raptor Codes," IEEE Transactions on Information Theory, Vol. 52, no. 6, pp. 2551-2567, 2006. Introduction [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. • 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 [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 : = +

  9. Full rank decoding • Whenever the ripple is empty • An early termination • A particular symbol is borrowed • Decoded through some other method • Placing the symbol into the ripple for the LT decoding process to continue. • 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 Wiedemannalgorithm 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 [I I] D. Wiedemann, "Solving sparse linear equations over finite fields," IEEE Transactions on Information Theory, Vol. 32, no. I, pp. 54-62, 1986. • 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]) [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. Random matrix rank when n=k • The state transition probability : • This allows us to determine the degree distribution of the sum of any number of vectors.

  20. Random matrix rank when n=k • 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. Random matrix rank when n=k • 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

  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.

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