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I do not understand how the constant map $x\mapsto a^8+a$ defines a hyperplane, but is 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 $K = GF(2^{n+1})$, 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 1 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)
]

I do not understand how the constant map $x\mapsto a^8+a$ defines a hyperplane, but is if you want to find a basis of the orthogonal hyperplane of the vector $a^8+a$, where $a$ is "the" generator of $K $F = GF(2^{n+1})$ viewed as a vector space over $K = GF(2^{n+1})$, 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 1 7 over Finite Field of size 2
sage: v = V(a^8+a) 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)
]

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 $F = GF(2^{n+1})$ viewed as a vector space over $K = GF(2^{n+1})$, 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)
]

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 $F $K = GF(2^{n+1})$ viewed as a vector space over $K $F = GF(2)$, 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)
]