# ApCoCoA-1:CharP.MBBasisF2

## CharP.MBBasis

Computing a Border Basis of a given ideal over F_2.

### Syntax

```CharP.MBBasisF2(F:LIST):LIST
CharP.MBBasisF2(F:LIST, NSol: INT):LIST
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Let f_1, ... , f_m is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by f_1, ... , f_m and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of F_2 rational solutions. The first version is safe to use if you do not know the exact number of F_2 rational solutions.

• @param F: List of polynomials.

• @param NSol: Number of F_2 rational solutions.

• @return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials.

#### Example

```Use Z/(2)[x[1..4]];
F:=[
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + x + x + 1,
xx + xx + xx + xx + 1
];

-- Then we compute a Border Basis with
CharP.MBBasisF2(F);

The size of Matrix is:
No. of Rows=4
No. of Columns=11
The size of Matrix is:
No. of Rows=8
No. of Columns=11
No. of mutants found =1
The size of Matrix is:
No. of Rows=11
No. of Columns=11
No. of mutants found =2
The size of Matrix is:
No. of Rows=16
No. of Columns=11
No. of mutants found =0
The size of Matrix is:
No. of Rows=31
No. of Columns=15
No. of mutants found =0

[x + 1, x, x + 1, x]

```

#### Example

```Use Z/(2)[x[1..4]];
F:=[
xx + xx + xx + xx + x + x + x + x,
xx + xx + xx + x + x + x,
xx + xx + xx + xx + x + x,
xx + xx + xx + xx + x + x
];

NSol:=3;

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions
-- Compute the solution with
CharP.MBBasisF2(F,NSol);

The size of Matrix is:
No. of Rows=4
No. of Columns=9
The size of Matrix is:
No. of Rows=14
No. of Columns=14
The size of Matrix is:
No. of Rows=16
No. of Columns=15
[xx + x, xx + x, xx + x, xx + x, xxx + x, xxx + x]

```