subset (k),and defines a subset Gck)=Π光 of the corresponding local Cartesian product set).The local constraint thus defines a set of valid local configurations ("local codewords")xiI(k)C,where the notation xdenotes the projection of a configuration x onto the symbol variables indexed by(k) The code Cis then the set of all configurations that satisfy all local constraints C=xexx)C for all kek) For example,a linear code Cmay be characterized as the set of xe&that satisfy the parity-check equations <x,h>=0 for a certain set of check configurations (h.k).The symbol variables X,that are involved in the kth check are those for which h0.Each local code C is then a linear (m.n-1,2)single-parity-check (SPC)code, whose length nis equal to the number of symbols involved in C A behavioral realization has a nature graphical model,which in coding theory is called a Tanner graph. 4.6.3 Tanner Graph The above elementary linear behavioral realizations can be represented conveniently by a graphica Graph Notation A graph G is a triple consisting of a vertex set V(G),an edge set E(G),and a relation that associate with each edge two vertices called its endpoints. If vertex veV is an endpoint of edge eE,then v and e are incident.The degree of a vertex veVis the number of incident edges and will be denoted by d(v). A graph G is called regular when all of its vertices have degree r,i.e., v∈'：d(v)=r Bipartite Graph:A bipartite graph is a graph whose vertices may be partitioned into two sets,and where edges may only connect two vertices not residing in the same set Amore precise definition is given below. 定义：若图G的节点集P可以划分为两个子集Y和'2：VU乃='，∩=p,使得 3 3 subset I(k), and defines a subset ( ) ( ) k i i k k ∈ ⊆ = ∏ I CX X of the corresponding local Cartesian product set X(k).The local constraint thus defines a set of valid local configurations (“local codewords”) x|() I ki k = {xi k , () ∈ ∈ I C } , where the notation x|I(k) denotes the projection of a configuration x onto the symbol variables Xi indexed by I(k). The code C is then the set of all configurations that satisfy all local constraints CX C K =∈ ∈ ∈ {x x| for all | () I k k k } For example, a linear code C ⊆X may be characterized as the set of x∈X that satisfy the parity-check equations , 0 < >= k x h for a certain set of check configurations { , } k h ∈ ∈ X K k . The symbol variables Xi that are involved in the kth check are those for which hki ≠0. Each local code Ck is then a linear (nk, nk-1, 2) single-parity-check (SPC) code, whose length nk is equal to the number of symbols involved in Ck. A behavioral realization has a nature graphical model, which in coding theory is called a Tanner graph. 4.6.3 Tanner Graph The above elementary linear behavioral realizations can be represented conveniently by a graphical model called Tanner graph. Graph Notation „ A graph G is a triple consisting of a vertex set V(G), an edge set E(G), and a relation Φ that associate with each edge two vertices called its endpoints. If vertex v∈V is an endpoint of edge e∈E, then v and e are incident. The degree of a vertex v∈V is the number of incident edges and will be denoted by d(v) . „ A graph G is called regular when all of its vertices have degree r; i.e., ∀∈ = v V dv r : ( ) . „ Bipartite Graph: A bipartite graph is a graph whose vertices may be partitioned into two sets, and where edges may only connect two vertices not residing in the same set. A more precise definition is given below. 定义: 若图 G 的节点集V 可以划分为两个子集V1和V2 ： 12 12 V V VV V ∪= ∩= , φ ,使得