Lt codes decoding design and analysis
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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


Lt codes decoding design and analysis

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


Full rank decoding2

Full rank decoding


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


Random matrix rank when n k7

Random matrix rank when n > k


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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