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where x0 , x1 , . . ., xn 1 are the n points selected for evaluation of the polynomial. They can be chosen arbitrarily, for example, for simplicity of encoding, or alternatively they can be all the possible integer values that can be represented over the number of bits available. The encoding matrix of Reed Solomon codes has a special form characterized by a geometric progression in each row. Such matrices are named Vandermonde matrices. Decoding Process of RS-Codes. The decoding process consists in reconstructing all the polynomial coef cients a0 , a1 , . . . , ak 1 in a unique way. As is commonly known, for this to be possible, it is suf cient to know the value of the polynomial at exactly k points; that is, receiving k code-words is suf cient. Hence, assuming that the identity (e.g., the sequence number) of the code-words that have arrived to destination is known, the coef cients of the polynomial can be derived at the destination site by solving the following system of equations. yj,1 1 ... = ... ... ... yl,k 1 1 yi,0 1 xi,0 xj,1 ... ... xl,k 1

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template <int hi, int wid> class Screen { public: Screen() : _height( hi ), _width( wid ), _cursor ( 0 ), _screen( hi * wid, `#' ) { } // private: string _screen;

file:///F|/WinDDK/resources/CPPPrimer/c++primerhtm (751 / 1065) [2001-3-29 11:32:12]

string::size_type _cursor; short _height; short _width; }; typedef Screen<24,80> termScreen; termScreen hp2621; Screen<8,24> ancientScreen;

log_p-bit subtractors r 1,2,3

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The expression to which a nontype parameter is bound must be a constant expression That is, it must be possible to evaluate it at compile-time In the previous example, the typedef termScreen refers to the template instantiation Screen<24, 80> The template argument for hi is 24 and the argument for wid is 80 In both cases the template argument is a constant expression However, given the class template BufPtr defined here, its instantiation results in a compiler error since the pointer value resulting from the invocation of operator new() is not known until run-time:

2 xi,0 2 xj,1

template <int *ptr> class BufPtr { }; // error: template argument cannot be evaluated at compile-time BufPtr< new int[24] > bp;

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Similarly, the value of a non-const object is not a constant expression It cannot be used as a template argument for a nontype template parameter However, the address of any object of namespace scope (even if the object is not of const type) is a constant expression (whereas the address of a local object is not) The address of an object of namespace scope can therefore be used as the argument for a nontype template parameter Similarly, the result of a sizeof expression is a constant expression that can be used as the argument for a nontype template parameter:

template <int size> Buf{ }; template <int *ptr> class BufPtr { }; int size_val = 1024; const int c_size_val = 1024; Buf< 1024 > buf0; // ok Buf< c_size_val > buf1; // ok Buf< sizeof(size_val) > buf2; // ok: sizeof(int) BufPtr< &size_val > bp0; // ok // error: cannot be evaluated at compile-time Buf< size_val > buf3;

Here is another example that illustrates how a nontype template parameter can be used to represent a constant value in a class template definition, and how a template argument is used to specify a value for this template parameter:

... ...

template <class Type, int size> class Fixed_Array { public: Fixed_Array( Type *ar) : count( size ) {

file:///F|/WinDDK/resources/CPPPrimer/c++primerhtm (752 / 1065) [2001-3-29 11:32:12]

for ( int ix = 0; ix < size; ++ix ) array[ ix ] = ar[ ix ]; } private: Type array[ size ]; int count; }; int ia[4] = { 0, 1, 2, 3 }; Fixed_Array< int, sizeof( ia ) / sizeof( int ) > iA( ia );

Expressions that evaluate to the same value are considered equivalent template arguments for a template nontype parameter The following three Screen instances, for example, all refer to the same template instantiation Screen<24,80>:

deg(v)

k 1 . . . xi,0

const int width = 24; const int height = 80; // all: type Screen< 24, 80 > Screen< 2*12, 40*2 > scr0; Screen< 6+6+6+6, 20*2+40 > scr1; Screen< width, height > scr2;

Some conversions are allowed between the type of a template argument and the type of a nontype template parameter The set of conversions allowed is a subset of the conversions allowed on function arguments: 1 Lvalue transformations, including lvalue-to-rvalue conversion, array-to-pointer conversion, and function-to-pointer conversion; for example:

template <int *ptr> class BufPtr { }; int array[10]; BufPtr< array > bpObj; // array-to-pointer conversion

2 Qualification conversions; for example:

template <const int *ptr> class Ptr { }; int iObj; Ptr< &iObj > pObj; // conversion from int* to const int*

The matrix utilized in the decoding process is a submatrix of the encoding matrix and is obtained by selecting the k rows that correspond to the arrived code-words (in the example the ith, jth, . . . , and the lth rows have been selected). The system admits a solution if the matrix is nonsingular. The determinant of the above k k

3 Promotions; for example:

template <int hi, int wid> class Screen { }; const short shi = 40; const short swi = 132; Screen< shi, swi > bpObj2; // promotion from short to int

file:///F|/WinDDK/resources/CPPPrimer/c++primerhtm (753 / 1065) [2001-3-29 11:32:12]

4 Integral conversions; for example:

Inverter in GF(pn).

Vandermonde matrix corresponds to the following expression. det(V ) =