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neighbors of an encoded symbol are those symbols of the previous layer of the bipartite graph to which the encoded symbol is connected by edges. An encoded symbol is obtained by XOR-ing over all its neighbors of the previous graph layer.

All that remains to be done now is to initialize the static members free_list and QueueItem_chunk The template form for the definition of a static data member is as follows:

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/* for each QueueItem instantiation, generate the * associated free_list, initialize it to 0 */ template <class T> QueueItem<T> *QueueItem<T>::free_list = 0; /* for each QueueItem instantiation, generate the * associated QueueItem_chunk, initialize it to 24 */ template <class T> const unsigned int QueueItem<T>::QueueItem_chunk = 24;

The template definition of a static data member must appear outside the class template definition For this reason, the template definition starts with the keyword template followed by the class template parameter list <class T> The name of the static data member is prefixed by QueueItem<T>::, which indicates that the member belongs to the class template QueueItem The static data member definitions are added to the header file Queueh These definitions must be included in the files in which instantiations of the static data members are used (We look at why we decided to do this, and at the related topic of template compilation model, in Section 168) A static data member is instantiated from the template definition only if it is used in a program A static member of a class template is itself a template The template definition for the static data member does not cause any memory to be allocated Memory is only allocated for particular instantiations of the static data member Each static data member instantiation corresponds to a class template instantiation An instantiation of a static data member, then, is always referred to through a particular class template instantiation For example:

QueueItem is not an actual instantiation = QueueItem::QueueItem_chunk; = QueueItem<string>::QueueItem_chunk; // ok = QueueItem<int>::QueueItem_chunk; // ok

1. Find an output symbol yi that is connected to only one input symbol xj . If there is no such output symbol, the decoding algorithm halts at this point and fails to recover all the input symbols. 2. Set xj = yi . 3. Add (i.e., perform a bit-by-bit XOR) xj to all the output symbols that are connected to xj . 4. Remove all the edges connected to the input symbol xj . 5. Repeat until all {xj } are determined, or otherwise no more output symbols can be found that have exactly one neighbor. In the rst case, decoding ends successfully; in the latter, it fails. LT codes require a simple decoder, but it turns out that the degree distribution is a critical part of the design. The decoding process as described above will not even start if there is no output symbol of degree 1. This means that a good degree distribution is required for good decoding performances. Indeed, whether or not the decoding algorithm is successful depends solely on the degree distribution. The degree distribution used to produce LT codes is the Soliton Distribution or the more ef cient Robust Soliton Distribution. Further details on the degree distribution analysis and the Soliton and Robust Soliton distributions can be found in Luby [28].

Exercise 167 Using operator new(), operator delete(), and their associated static members screenChunk and freeStore defined in Section 158, implement these operators and static members for the class template Screen defined in Exercise 166

The class template QueueItem is designed to serve only in the implementation of a Queue To enforce this, QueueItem has a private constructor that allows the member functions of its friend class template Queue, but no other classes or functions (except its own member functions), to create objects of the type QueueItem Although QueueItem is a class template visible to the entire program, the program is unable either to create QueueItem objects or to refer to any of QueueItem's members without calling Queue's member functions An alternative implementation strategy is to nest the definition of the class template QueueItem within the private section of the class template Queue With QueueItem being a nested private type, it becomes inaccessible to the general program And because it is a private nested type, only the class template Queue and the friends of Queue the output operator can access it If we make the members of QueueItem public, it no longer becomes necessary to declare Queue as a friend to QueueItem

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mul_done first 1 exp_in 0 first s last s a exponentiator mul_start by_one exp_out enable control unit