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level of check bits is complete whereas the next-to-last level as well as all the previous ones still need reconstruction The decoding process proceeds one level at a time and requires nding each time a check bit on the right side such that only one adjacent message bit (among those that have generated it) is missing The missing message bit can thus be obtained by XOR-ing among the check bit and its known message bits As an example (see Figure 46b), suppose that c is a check bit whereas x1 , x2 , and x3 are the message bits it depends on, ie, c = x1 x2 x3 Suppose also that, after transmission, c, x1 , and x2 are known whereas x3 is unknown Then, the following holds true x1 x2 c = (x1 x2 ) (x1 x2 ) x3 = x3 .

That done, each time a Queue instantiation is passed to an ostream, the function template is instantiated and invoked Here is one possible implementation of the output operator as a function template:

template <class Type> ostream& operator ( ostream &os, const Queue<Type> &q ) { os "< "; QueueItem<Type> *p; for ( p = qfront; p; p = p->next ) os *p " "; os " >"; return os; }

If a Queue of objects of type int contains the values 3, 5, 8, and 13, the output of this Queue displays as follows:

Notice that the output operator refers to Queue's private member front The next thing we need to do is to declare the operator as a friend of Queue, as follows:

b(2)

The missing message bits are yielded one at a time by exploiting the graph relations in which they are involved together with other known message bits, to produce other known check bits Once decoded a single level, the previous one is then decoded By repeating the same procedure for each bipartite graph, the original k-bit message can nally be reconstructed, as well Properties of Tornado Codes This subsection provides some insights into interesting features of Tornado codes related to constraints and requirements of the encoding process, and it also discusses some advantages and drawbacks Property 1 The rst property is about the feasibility of Tornado codes This is strictly connected to the particular characteristics of the multi-bipartite graph in that successful decoding is only possible if an appropriate degree distribution is ful lled during construction of the graph.

template <lass Type> class Queue { friend ostream& operator ( ostream &, const Queue<Type> & ); // };

Notice that, in this case, as with the friend declaration within the class template Queue earlier in this section, this declaration creates a one-to-one mapping between an instantiation of Queue and its corresponding operator () instance

file:///F|/WinDDK/resources/CPPPrimer/c++primerhtm (762 / 1065) [2001-3-29 11:32:12]

The actual display of the Queue's elements relies on QueueItem's output operator ():

The degree of a node belonging to the kth graph level gives the number of nodes of the (k 1)th level that this node is generated by Luby et al [27] make use of linear programming techniques to select the degrees to assign to the nodes of the same level and speci cally de ne appropriate sequences of node degrees Property 2 The shape of the multi-bipartite graph also impacts the performance of Tornado codes Both encoding and decoding times strictly depend on the number of edges which are present in the multi-bipartite graph Clearly, the higher the number of edges, the higher the computational burden of the code The aforementioned encoding approach based on linear programming converges in linear time and also allows linear time decoding, as has been formally demonstrated by Luby et al [27].

os *p;

The QueueItem output operator also needs to be implemented as a template function This ensures that an appropriate operator () is instantiated automatically when needed:

template <class Type> ostream& operator ( ostream &os, const QueueItem<Type> &qi ) { os qiitem; return os; }

Because this operator accesses QueueItem's private member item, this operator must be declared as a friend of the class template QueueItem This is done as follows:

template <class Type> class QueueItem { friend class Queue<Type>; friend ostream& operator ( ostream &, const QueueItem<Type> & ); // };