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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

LT Codes Decoding: Design and Analysis

Feng Lu

Chuan HengFoh, JianfeiCai and Liang- TienChia

Information Theory, 2009. ISIT 2009. IEEE International Symposium on 

outline
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
introduction

[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

introduction1
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
full rank lt decoding process
Full rank LT decoding process
  • LT decoding drawbacks
  • Full rank decoding
  • Recovering the borrowed symbol
  • Non-square case
lt decoding drawbacks
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.
lt decoding drawbacks1
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
slide8
GF(2)
  • GF(2) is the Galois field of two elements.
  • The two elements are nearly always 0 and 1.
  • Addition operation :
  • Multiplication operation :

=

+

full rank decoding
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
full rank decoding1
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
recovering the borrowed symbol
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

=

recovering the borrowed symbol1
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.
recovering the borrowed symbol2
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.

non square case
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
random matrix rank
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.

random matrix rank when n k
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
random matrix rank when n k1
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
random matrix rank when n k2
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.
random matrix rank when n k3
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)
random matrix rank when n k4
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
random matrix rank when n k5
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
random matrix rank when n k6
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

numerical results and discussion
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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