Feng Lu Chuan Heng Foh, Jianfei Cai and Liang- Tien Chia
Information Theory, 2009. ISIT 2009. IEEE International Symposium
on LT Codes Decoding: Design and Analysis
Slide 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
Slide 3
Introduction 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 k10 [9] using Gaussian elimination method for decoding The
decoding complexity increase significantly [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
Slide 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
Slide 5
Full rank LT decoding process LT decoding drawbacks Full rank
decoding Recovering the borrowed symbol Non-square case
Slide 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.
Slide 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
Slide 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 : +01 00 1 110 01 000 101
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Full rank decoding 1. Whenever the ripple is empty An early
termination 2. A particular symbol is borrowed Decoded through some
other method 3. Placing the symbol into the ripple for the LT
decoding process to continue. 4. 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
Slide 10
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
Slide 11
Full rank decoding
Slide 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
Slide 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.
Slide 14
Recovering the borrowed symbol 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]) [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.
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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
Slide 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.
Slide 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
Slide 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
Slide 19
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
Slide 20
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)
Slide 21
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
Slide 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
Slide 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
Slide 24
Random matrix rank when n > k
Slide 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.