Find the irreducible factors of the polynomial \(T^9 - 1\) in \(\mathbb{F}_7[T]\).
(You should include proofs that the factors you describe are irreducible).
Let \(0 < k,m \in \mathbb{N}\), put \(n =mk\), and consider the subspace \(C \subset \mathbb{F}_q^n\) defined by \[C = \{ (v,v,\cdots,v ) \mid v \in \mathbb{F}_q^k\} \subset \mathbb{F}_q^n.\] Find the minimal distance \(d\) of this code.
For example, if \(n = 6\), \(k=3\) and \(m = 2\) then \[C = \{(a_1,a_2,a_3,a_1,a_2,a_3) \mid a_i \in \mathbb{F}_q\} \subset \mathbb{F}_q^6.\]
(Corrected)
By an \([n,k,d]_q\)-system we mean a pair \((V,\mathcal{P})\), where \(V\) is a finite dimensional vector space over \(\mathbb{F}_q\) and \(\mathcal{P}\) is an ordered finite family \[\mathcal{P} = (P_1,P_2,\cdots,P_n)\] of points in \(V\) (in general, points of \(\mathcal{P}\) need not be distinct – you should view \(\mathcal{P}\) as a list of points which may contain repetitions) such that \(\mathcal{P}\) spans \(V\) as a vector space. Evidently \(|\mathcal{P}| \ge \dim V\).
The parameters \([n,k,d]\) are defined by \[n = |\mathcal{P}|, \quad k = \dim V, \quad d = n - \max_H |\mathcal{P} \cap H|.\] where the maximum defining \(d\) is taken over all linear hyperplanes \(H \subset V\) and where points are counted with their multiplicity – i.e. \(|\mathcal{P} \cap H| = |\{i \mid P_i \in H \}|\).
Gjven a \([n,k,d]_q\)-system \((V,\mathcal{P})\), let \(V^*\) denote the dual space to \(V\) and consider the linear mapping \[\Phi:V^* \to \mathbb{F}_q^n\] defined by \[\Phi(\psi) = (\psi(P_1),\cdots,\psi(P_n)).\]
Show that \(\Phi\) is injective.
Write \(C = \Phi(V^*)\) for the image of \(\Phi\), so that \(C\) is an \([n,k]_q\)-code. Show that the minimal distance of the code \(C\) is given by \(d\).
Conversely, let \(C \subset \mathbb{F}_q^n\) be an \([n,k,d]_q\)-code, and put \(V = C^*\). Let \(e^1,\cdots,e^n \in (\mathbb{F}_q^n)^*\) be the dual basis to the standard basis. The restriction of \(e^i\) to the subspace \(C\) determines an element \(P_i\) of \(C^* = V\). Write \(\mathcal{P} = (P_1,P_2,\cdots,P_n)\) for the resulting list of vectors in \(V\)..
Prove that the minimum distance \(d\) of the code \(C\) satisfies \[d = n - \max_H | \mathcal{P} \cap H |.\]
Let \(C\) be the linear code over \(\mathbb{F}_5\) generated by the matrix \[G = \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 2 \\ 0 & 1 & 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 & 1 & 1 \end{pmatrix}.\]
Find a check matrix \(H\) for \(C\).
Find the minimum distance of \(C\).
Decode the received vectors \((0,2,3,4,3,2)\) and \((0,1,2,0,4,0)\) using syndrome decoding.