因子圖(factor graph)是一種由兩個部分構成的圖形模型，意思就是因子圖中含有兩種節點－變數節點及函式節點；因子圖是用來表示一個總體函式(global function)如何被因式分解成許多局部函式(local function)的乘積，而總體函數中每一個變數和每一個局部函式都有其相對應的變數節點和函式節點，而和積演算法(sum-product algorithm)是一個能有效運作於因子圖上的演算法。當我們應用因子圖來表示錯誤更正碼時，將和積演算法運作於因子圖上時，其效應便等同於許多已存在的解碼演算法，像是前後向演算法(forward/backward algorithm)、裴特比演算法(Viterbi algorithm)，均可被看做是和積演算法的特例。不論是否有循環存在於因子圖中，和積演算法均可有效地在因子圖上運作；當因子圖沒有循環時，用任何一種排程法的和積演算法終究會跑到一個終點，且所得到的最終結果均相同，而得以正確計算出每一個邊際函式(marginal function)，這裡所指的終點是存在於所有邊上之訊息(message)都不會再更動；但當因子圖有循環時，就無法正確計算邊際函式，且不能保證排程法對於結果會否有影響；和積演算法可以無止盡地運作於有循環的因子圖上，其效果就像是重算性解碼演算法(iterative decoding algorithm)。包含渦輪解碼(turbo decoding)在內，傳統的重算性演算法可被看作是利用平行排程法(parallel scheduling)之和積演算法的一個特例，但我們知道有時它會發散。從實驗觀察的結果，我們認為傳統的重算性演算法之所以會發散，是由於其排程方式是決定性的(deterministic)；故本篇論文的主旨，是提出一個隨機的排程方式以解決發散性的問題，命名為馬可夫排程(Markovian scheduling)。

Factor graphs are bipartite graphs used for expressing how a global function factors into a product of local functions. In
this thesis, we focus on factor graphs of codes. Some decoding algorithms, such as the BCJR algorithm, the Viterbi

algorithm, and the Pearl's belief propagation algorithm, can be treated as instances of the sum-product algorithm applied to

factor graphs of codes. The sum-product algorithm can be applied to an arbitrary factor graph, whether it is cycle-free or

not. When a factor graph of a code is cycle-free, the sum-product algorithm will be terminated and accurately compute all

marginal functions of the global function. But if the graph is loopy, the sum-product algorithm can be run endlessly and

results in an iterative decoding algorithm. The traditional iterative decoding algorithm is an instance of the sum-product

algorithm with parallel scheduling. Unfortunately, it sometimes diverges. In this thesis, we propose a stochastic sequential

scheduling scheme, called Markovian scheduling, to avoid the divergence behavior. Simulation-study shows that this

Markovian scheduling is successful.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, ``Optimal decoding of linear codes for minimizing symbol error rate'' IEEE
Trans. Inform. Theory, vol. IT-20, pp. 284--287, Mar. 1974.
G. Berrou, A. Glavieux, and P. Thitimajshima, ``Near Shannon limit error-correcting coding: turbo codes,'' in Proc. 1993
Int. Conf. Commun., Geneva, Switzerland, May 1993, pp. 1064--1070.
W. T. Freeman, and Y. Weiss, ``On the fixed points of the max-product algorithm,'' TR-99-39, Jan. 2000.
R. G. Gallager, ``Low-density parity-check codes,'' IRE Trans. Inform. Theory, vol. IT-8, pp. 21--28, Jan. 1962.
F. R. Kschischang, and B. J. Frey, ``Iterative decoding of compound codes by probability propagation in graphical
models,'' IEEE J. on Selected Areas in Commun., vol. 16, no. 2, Feb. 1998.
F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, ``Factor graphs and the sum-product algorithm,'' preprint.
R. J. McEliece, ``On the BCJR trellis for linear block codes,'' IEEE Trans. Inform. Theory, vol. 42, no. 4, Jul. 1996.
R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng, ``Turbo decoding as an instance of Pearl's `belief propagation' algorithm,'' IEEE J. on Selected Areas in Commun., vol. 16, pp. 140--152, 1998.
R. J. McEliece, E. R. Rodemich, and J.-F. Cheng, ``The turbo decision algorithm,'' Presented at the 33rd Allerton
Conference on Communication, Control and Computing, Oct. 1995.
R. M. Tanner, ``A recursive approach to low complexity codes,'' IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547,
Sept. 1981.
Y. Weiss, ``Belief propagation and revision in networks with loops,'' Technical Report 1616, MIT AI lab, 1997.
N. Wiberg, ``Codes and decoding on general graphs,'' PhD thesis, Linkoping University, Sweden, 1996.