Example CodeFld_CosetLeaders (H161E13)

We construct a Hamming code C, encode an information word using C, introduce one error, and then decode by calculating the syndrome of the "received" vector and applying the CosetLeaders map to the syndrome to recover the original vector.

First we set C to be the third order Hamming Code over the finite field with two elements.

> C := HammingCode(GF(2), 3);
> C;
[7, 4, 3] Hamming code (r = 3) over GF(2)
Generator matrix:
[1 0 0 0 0 1 1]
[0 1 0 0 1 0 1]
[0 0 1 0 1 1 0]
[0 0 0 1 1 1 1]

Then we set L to be the set of coset leaders of C in its ambient space V and f to be the map which maps the syndrome of a vector in V to its coset leader in L.

> L, f := CosetLeaders(C);
> L;
{@
    (0 0 0 0 0 0 0),
    (1 0 0 0 0 0 0),
    (0 1 0 0 0 0 0),
    (0 0 1 0 0 0 0),
    (0 0 0 1 0 0 0),
    (0 0 0 0 1 0 0),
    (0 0 0 0 0 1 0),
    (0 0 0 0 0 0 1)
@}

Since C has dimension 4, the degree of the information space I of C is 4. We set i to be an "information vector" of length 4 in I, and then encode i using C by setting w to be the product of i by the generator matrix of C.

> I := InformationSpace(C);
> I;
Full Vector space of degree 4 over GF(2)
> i := I ! [1, 0, 1, 1];
> w := i * GeneratorMatrix(C);
> w;
(1 0 1 1 0 1 0)

Now we set r to be the same as w but with an error in the 7-th coordinate (so r is the "received vector").

> r := w;
> r[7] := 1;
> r;
(1 0 1 1 0 1 1)

Finally we let s be the syndrome of r with respect to C, apply f to s to get the coset leader l, and subtract l from r to get the corrected vector v. Finding the coordinates of v with respect to the basis of C (the rows of the generator matrix of C) gives the original information vector.

> s := Syndrome(r, C);
> s;
(1 1 1)
> l := f(s);
> l;
(0 0 0 0 0 0 1)
> v := r - l;
> v;
(1 0 1 1 0 1 0)
> res := I ! Coordinates(C, v);
> res;
(1 0 1 1)