Iterative Soft-Decision Decoding of Algebraic- Geometric Codes Li Chen Associate Professor School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China [email protected]website: sist.sysu.edu.cn/~chenli Institute of Network Coding and Department of Information Engineering, the Chinese University of Hong Kong 1 st of Aug, 2012
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Iterative Soft-Decision Decoding of Algebraic-Geometric Codes Li Chen Associate Professor School of Information Science and Technology, Sun Yat-sen University,
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Iterative Soft-Decision Decoding of Algebraic-Geometric Codes
Li Chen Associate Professor School of Information Science and Technology,
Sun Yat-sen University, Guangzhou, China [email protected] website: sist.sysu.edu.cn/~chenli
Institute of Network Coding and Department of Information Engineering,
the Chinese University of Hong Kong 1st of Aug, 2012
Outline Introduction (How to construct an algebraic-geometric code?)
Review on Koetter-Vardy list decoding (Challenges in the decoding)
I. Introduction The construction of an algebraic-geometric (AG) code
Based on an algebraic curve χ(x, y, z) Identify its point of infinity p∞ define the pole basis Φ
Pick up one of the affine components, e.g., χ(x, y, 1) find out the affine points pj
The Reed-Solomon (RS) code is the simplest AG code Constructed based on y = 0; Its pole basis Φ = {1, x, x2, x3, x4, ……} Affine points {x1, x2, x3, …., xn} \ {0} Note: the length of the code cannot exceed the size of the finite field.
The generator matrix G The parity-check matrix H
I. Introduction The Hermitian curve: , ,
The point of infinity
The pole basis
Bivariate monomials and their pole orders
Based on one of its affine components Hw(x, y, 1), determine the affine points pj = (xj, yj, 1) where xj
w+1 + yjw + yj = 0 and j = 1, 2, …, n.
Encoding of an (n, k) Hermitian code Given the message vector The codeword is generated by Note > q The length of the code can exceed the size of the finite field!
I. Introduction Example: Construction of a (8, 4) Hermitian code
Defined in = {0, 1, α, α2}; The Hermitian curve H2(x, y, z) = x3 + y2z + yz2 point of infinity p∞ = (0, 1, 0);
One of its affine components: H2(x, y, 1) = x3 + y2 + y; Its pole basis Its affine points: p1 = (0, 0), p2 = (0, 1), p3 = (1, α), p4 = (1, α2),
III. Iterative Soft-Decision Decoding Based on Hb’, perform 3 BP iterations, we have the updated LLR vector as
The updated reliability matrix Π’ becomes
• For the ‘wrong’ LLR values ( ): we would like to change its sign, or reduce its magnitude;• For the ‘right’ LLR values: we would like to leave the sign unchanged and increase its magnitude;
= 4.478
= 4.037
Based on Theorem 1, KV decoding will succeed!
III. Iterative Soft-Decision Decoding Why Gaussian elimination should be bit reliability oriented?
reliable bitsunreliable bits
L’(c7)L’(c5)
Tanner graph
4/1 5/2 5/2 3/2 3/2 5/0
III. Iterative Soft-Decision Decoding How to improve the iterative decoding performance? It is possible that reliable bits are wrongly estimated by their LLR values; We can create different sets of bit indices B and let more bits’
corresponding cols. also fall into the identity submatrix of Hb’.
• Note if there are multiple matrix adaptations, the next bit reliability sorting will be performed based on the updated LLR vector ;
• Multiple attempts of KV decoding result in an output list that contains all the message candidates. The Maximum Likelihood (ML) criterion is used to select one from the list.
IV. Geometric Interpretation Insight of why we need matrix adaptations before the BP decoding Normalize the vector to the vector Normalize L(cj) to Tj by the mapping function
A graphical look into the vector and the vector.
IV. Geometric InterpretationWhen the codeword is not found When the codeword is found
• When a codeword is found, Tj = 1 for j = 1, 2, …, N;
IV. Geometric Interpretation Objective of the BP decoding: Finding the vector that minimizes the potential
function
The LLR update in the BP decoding
can be seen as the T value update
Finding the estimated codeword using the BP algorithm can be seen as identifying the vertex at which the potential function is minimized.
IV. Geometric Interpretation The convergence behavior of the potential function of the (64, 39) Hermitian code = -100
V. Complexity Reduction Decoding parameters -- number of groups of unreliable bit indices -- number of matrix adaptations (Gau. eliminations) -- number of BP iterations
There are three types of computations required by the decoding Binary operations (Gau. eliminations): Floating point operations (BP iterations): Finite field arithmetic operations (KV decodings):
With the iterative decoding parameters of Binary operations: Floating point operations: Finite field arithmetic operations:
×
×
×
V. Complexity Reduction Reduce the deployment of the KV decoding steps
ABP-KV decoding block diagram
We could try to assess the quality of matrices Π’ and M. If they are not good enough to result in a possibly successful decoding, the following KV decoding process will NOT be carried out.
ABP Π’MΠ Π’
Intp. Fac.M
V. Complexity Reduction Reliability-based received word score Multiplicity-based received word score Example: the (8, 4) Hermitian code
0
1
α
α2
0
1
α
α2
= 6.7
= 15
V. Complexity Reduction Recall the two theorems for successful KV decoding Theorem 1 If ( ) {KV can succeed;} Theorem 2 If ( ) {KV can succeed;}
Lemma 3 If ( ) {KV cannot succeed;}
Lemma 4 If ( ) {KV cannot succeed;}
ABP Π’MΠ Π’
Intp. Fac.M
Proof:
Proof:
V. Complexity Reduction
Complexity reduction for ABP-KV decoding of the (64, 52) Hermitian code Decoding parameters = (10, 5, 2) There are 50 KV decoding processes for each codeword frame
V. Complexity Reduction Other facilitated decoding approaches: Parallel decoding:
Output validation: once is found, the iterative decoding will be terminated.
( ; )
VI. Performance Analysis Decoding parameters: the KV decoding output list size (l) and The (8, 4) Hermitian code over the AWGN channel
VI. Performance Analysis The (64, 39) Hermitian code over the AWGN channel
VI. Performance Analysis The (64, 47) Hermitian code over the AWGN channel
VI. Performance Analysis The (64, 47) Hermitian code over the fast Rayleigh fading channel Coherent detection with the knowledge of CSI
VI. Performance Analysis Herm. (64, 47) vs. RS (15, 11), over the AWGN channel
VII. Conclusions Revisit the construction of AG codes: pole basis + affine points;
Review the KV soft-decision list decoding algorithm: Π dependent;
Introduce an iterative soft-decision decoding algorithm for Hermitian codes:
Adaptive Belief Propagation + KV list decoding;
ABP algorithm is bit reliability oriented BP is also good for AG (RS) codes;
Geometric interpretation necessity of performing parity-check matrix adaptation;
Complexity reduction : successive criteria to assess Π’ and M; parallel decoding; output validations;
Performance analysis shows a significant performance gain can be achieved
(~ conventional algorithms; ~ RS codes).
Acknowledgement
Project: Advanced coding technology for future storage devices;