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### LT Codes Decoding: Design and Analysis

Feng Lu

Chuan HengFoh, JianfeiCai and Liang- TienChia

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

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

[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

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

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

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 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

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

- 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 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

- 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 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 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

- 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

- 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

- 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=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=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=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=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 > 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 > 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

[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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