# a bug in computing minimum distance, coding theory

For anyone who is familiar with coding theory, we build codes as a subspace of the vector space F_q^n from generating matrices, with all linear combinations of rows of this matrix. For any generating matrix defined over a Finite Field, SAGE has spesific attributes "C =LinearCode(G)" which constructs the code from generating matrix and "C.minimum_distance()" which computes the minimum distance of the code. However, there is probably a bug or something which I couldn't find out.

For example see this:

This code is a [7,3] linear code over the field GF(4,"a"), which has min. distance 3,(as computed with MAGMA), but SAGE computes its min. distance as 5, which is above upper bound!

This is the SAGE input-output:

```
K = GF(4,"a"); a = K.gen(); G = Matrix([[a, a + 1, 1, a + 1, 1, 0, 0], [0, a, a + 1, 1, a + 1, 1, 0], [0, 0, a, a + 1, 1, a + 1, 1], [a + 1, 0, 1, 0, a + 1, 1, a + 1], [a, a + 1, a + 1, 0, 0, a + 1, 1], [a + 1, a, a, 1, 0, 0, a + 1], [a, a + 1, 1, a + 1, 1, 0, 0]]); C=LinearCode(G); C; C.minimum_distance();
```

gives the output:

```
︡"Linear code of length 7, dimension 3 over Finite Field in a of size 2^2" "5"
```

However for the same conditions MAGMA says this:

```
︡> G6 := LinearCode <F,7|[[a, a + 1, 1, a + 1, 1, 0, 0], [0, a, a + 1, 1, a + 1, 1, 0], [0, 0, a, a + 1, 1, a + 1, 1], [a + 1, 0, 1, 0, a + 1, 1, a + 1], [a, a + 1, a + 1, 0, 0, a + 1, 1], [a + 1, a, a, 1, 0, 0, a + 1], [a, a + 1, 1, a + 1, 1, 0, 0]]>; G6;
```

[7, 3, 3] Linear Code over GF(2^2)

I don't understand where SAGE fails.. Do you have any idea?