Hamming codes

The $[7, 4, 3]$-Hamming code over $GF(2)$ having generator matrix

$$\left( \begin{array}{ccccccc} 1& 0& 0& 0& 0& 1& 1\\ 0& 1& 0... ... 0& 0& 1& 0& 1& 1& 0\\ 0& 0& 0& 1& 1& 1& 1 \end{array}\right)$$

is obtained by typing

C:=HammingCode(GF(2),3);
C;
G:=GeneratorMatrix(C);
G;

Encoding a message $w$ using $G$, is simply the map $w\longmapsto wG$. Type

W:=InformationSpace(C);
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); H; To see if a vector in $\mathbb{F}^7$ is a codeword, simply compute $Hv$ and check if it is zero or not. Here's a MAGMA example ¹:

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

Since this last vector is non-zero, $v$ 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 1 (a)   For the parity check matrix $H$ of the binary Hamming code of length $2^3-1=7$, verify $Hc=0$ for three or four codewords c. Decode $(1,1,0,0,0,0,0)$.

(b) Find a parity check matrix of the $3$-ary Hamming code of length $(3^3-1)/(3-1)=13$. Verify $Hc=0$ for three or four codewords $c$. Decode $(1,2,1,2,1,2,1,2,1,2,1,2,1)$.

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

Exercise 2   Find the dimension and minimum distance of

(a) the binary Hamming code of length $15$,

(b) the 3-ary Hamming code of length $13$.