## Reed-Muller codes

Reed-Muller shall be abbreviated RM.

The RM code over having generator matrix

is obtained by typing

C:=ReedMullerCode(1,3);
C:
G:=GeneratorMatrix(C);
G;

Encoding a message using , is simply the map . Type
W:=InformationSpace(C);
W;
Cwords:=<w*G : w in W>;
Cwords;
#Cwords;


From this, you see all the codewords of C and how many there are.

To get the parity check matrix, type H:=ParityCheckMatrix(C); To see if a vector in is a codeword, simply compute and check if it is zero or not. Here's a MAGMA example:

V:=AmbientSpace(C);
v:=V![1,0,0,0,0,0,0,0];
v*Transpose(H);


Since this last vector is non-zero, is not a codeword. If it was a vector received in transmission (with at least one error) then to decode it, hence to find the most likely codeword sent, type

Decode(C,v);


Exercise 3.15.4 (a)   For the parity check matrix of the RM code of length , verify for three or four codewords c. Decode .

(b) Find a parity check matrix of the RM code of length . Verify for three or four codewords . Decode .

To get the dimension of the code, type Dimension(C); To get its minimum distance, type MinimumDistance(C);

Exercise 3.15.5   Find the dimension and minimum distance of the RM code of length .

david joyner 2008-04-20