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
 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
 LT codes
 Large value of k :
Perform very well [5]
[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
 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
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 :
+
0
1
0
0
1
1
1
0
 Multiplication operation :
×
0
1
0
0
0
1
0
1
1 0
1 1
1 1
𝑦1
1 𝑥1
0 𝑥2 = 𝑦2
𝑦3
1 𝑥3
𝑥1 +𝑥3 = 𝑦1
Full rank decoding
Whenever the ripple is empty
1.

An early termination
A particular symbol is borrowed
2.

Decoded through some other method
Placing the symbol into the ripple for the LT decoding process to
continue.
4. Repeated until the LT decoding process terminates with a success
3.
 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 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
Full rank decoding
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
1 0
1 1
1 1
𝑦1
1 𝑥1
0 𝑥2 = 𝑦2
𝑦3
1 𝑥3
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
 We use the efficient Wiedemann algorithm [11] to solve
[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.
 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])
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
Initialized to
 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).
[15] R.A. Hom, C.R. Johnson, "Matrix Analysis," Cambridge
University Press, 1985
Random matrix rank when n > k
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.