Let \(q\) be a power of a prime \(p > 3\) and let \(k = \mathbb{F}_q\).
For a homogeneous polynomial \(F \in k[X,Y,Z,W]\), let us write \[V(F) = \{ P = (x:y:z:w) \in \mathbb{P}^3_k \mid F(x,y,z,w) = 0\}\] for the set of solutions of the equation \(F=0\) in \(\mathbb{P}^3_k\).
For \(a \in k^\times\), consider the polynomial \[F_a = XY + Z^2 - aW^2 \in k[X,Y,Z,W].\]
If \(4 \mid q -1\) show that \[|V(F_a)| = |V(X^2 + Y^2 + Z^2 - aW^2)|\]
Hint: First show that \(X^2 + Y^2 + Z^2 - aW^2\) is obtained from \(F_a\) by a linear change of variables.
If \(a = 1\), show that \(|V(F_1)| = q^2 + 2q + 1\).
Hint: Making a linear change of variables, first show that \(|V(F_1)| = |V(G)|\) where \(G = XY + ZW\).
To count the points \((x:y:z:w)\) in \(V(G)\), first count the points with \(xy = 0\) (and hence also \(zw = 0\)), and then the points with \(xy \ne 0\).
Let \(S = \{ a^2 \mid a \in k\}\).
Show that \(|S| = \dfrac{q+1}{2}\). Conclude that there are \(q - \dfrac{q+1}{2} = \dfrac{q-1}{2}\) non-squares in \(k\).
If \(a \in S\), show that \(|V(F_a)| = |V(F_1)| = q^2 + 2q + 1\).
If \(a \in k\), \(a \not \in S\), show for any \(\alpha \in k^\times\) that there are exactly \(q+1\) pairs \((c,d) \in k \times k\) with \(c^2 - ad^2 = \alpha\).
Hint: We may identify \(\ell = \mathbb{F}_{q^2} = \mathbb{F}_q[\sqrt{a}]\). Under this identification, the norm homomorphism \(N=N_{\ell/k}: \ell^\times \to k^\times\) is given by the formula \[N(c + d\sqrt{a}) = (c+d\sqrt{a})(c-d\sqrt{a}) = c^2 - ad^2.\] On the other hand, by Galois Theory, we have \(N(x) = x \cdot x^q = x^{1+q}\) for any \(x \in \ell\). Thus \(N(\ell^\times) = k^\times\) and \(|\ker N| = q+1\).
If \(a \in k\), \(a \not \in S\) show that \(|V(F_a)| = q^2 + 1\)
Hint: Notice that the equation \(Z^2 - aW^2 = 0\) has no solutions \((z:w) \in \mathbb{P}^1_k\), and use (e) to help count.
Let \(f = T^{11} - 1 \in \mathbb{F}_4[T]\).
Show that \(T^{11} -1\) has a root in \(\mathbb{F}_{4^5}\).
If \(\alpha \in F_{4^5}\) is a primitive element – i.e. an element of order \(4^5 -1\), find an element \(a = \alpha^i \in \mathbb{F}_{4^5}\) of order \(11\), for a suitable \(i\).
Show that the minimal polynomial \(g\) of \(a\) over \(\mathbb{F}_4\) has degree 5, and that the roots of \(g\) are powers of \(a\). Which powers?
Show that \(f = g\cdot h \cdot (T-1)\) for another irreducible polynomial \(h \in \mathbb{F}_4[T]\) of degree 5. The roots of \(h\) are again powers of \(a\). Which powers?
Show that \(\langle f \rangle\) is a \([11,6,d]_4\) code for which \(d \ge 4\).
Consider the following variant of a Reed-Solomon code: let \(\mathcal{P} \subset \mathbb{F}_q\) be a subset with \(n = |\mathcal{P}|\) and write \(\mathcal{P} = \{a_1,\cdots,a_n\}\).
Let \(1 \le k \le n\) and write \(\mathbb{F}_q[T]_{< k}\) for the space of polynomial of degree \(< k\), and let
\(C \subset \mathbb{F}_q^n\) be given by \[C = \{ (p(a_1),\cdots,p(a_n)) \mid p \in \mathbb{F}_q[T]_{<k}.\]
If \(n \ge k\), prove that \(C\) is a \([n,k,n-k+1]_q\)-code.
If \(P = \mathbb{F}_q^\times\), prove that \(C\) is a cyclic code.
If \(q = p\) is prime and if \(P = \mathbb{F}_p\), prove that \(C\) is a cyclic code.