# basis of hyperplane

Hallo

I am new to sage and have this problem.

Given a hyperplane $H_u \subset GF(2^{n+1})$ define by $x |-> a^8 +a$

How do I determine the basis of $H_u$ over $GF(2)$?

Regards

basis of hyperplane

Hallo

I am new to sage and have this problem.

Given a hyperplane $H_u \subset GF(2^{n+1})$ define by $x |-> a^8 +a$

How do I determine the basis of $H_u$ over $GF(2)$?

Regards

add a comment

1

I do not understand how the constant map $x\mapsto a^8+a$ defines a hyperplane, but if you want to find a basis of the orthogonal hyperplane of the vector $a^8+a$, where $a$ is "the" generator of $K = GF(2^{n+1})$ viewed as a vector space over $F = GF(2)$, you can:

```
sage: n = 6
sage: K = GF(2^(n+1),'a') ; K
Finite Field in a of size 2^7
sage: a = K.gen()
sage: F = K.base() ; F
Finite Field of size 2
sage: V = K.vector_space() ; V
Vector space of dimension 7 over Finite Field of size 2
sage: v = V(a^8 + a) ; v
(0, 0, 1, 0, 0, 0, 0)
sage: m = matrix(v) ; m
[0 0 1 0 0 0 0]
sage: m.right_kernel().basis()
[
(1, 0, 0, 0, 0, 0, 0),
(0, 1, 0, 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)
]
```

@tmonteil, thanx the map was suppose to be $x\mapsto x\cdot a^8 + x^8 \cdot a$. By adding the code f = lambda x: x * a^8 + a * x^8 S = V.subspace([f(x) for x in K]) S.basis() I got a solution. What will the most effective method to create a map (isomorphism) from $S$ to $GF(2^n)$?

Asked: **
2013-07-20 09:40:54 -0500
**

Seen: **638 times**

Last updated: **Jul 20 '13**

accessing the components of a vector

Linear Transformation Matrix is Transposed

How to show the steps of Gauss' method

How does one detect cyclic vectors in SAGE?

Row reduction modulo prime powers

Yet another linear combination

finding specific vectors froma parallelogram [closed]

Turning system of linear equations into a matrix

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.