We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic
trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$
determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over
$\mathbb{F}_{q^m}$, and three related codes $\mathcal{C}_{\mathcal{L},0}$,
$\mathcal{C}_{\mathcal{L},1}$ and $\mathcal{C}_{\mathcal{L},2}$. We describe
the spectra for all these codes when $\mathcal{L}$ is an even rank family, in
terms of the distribution of ranks of the forms $Q_R$ in the family
$\mathcal{L}$, and we also compute the complete weight enumerator for
$\mathcal{C}_\mathcal{L}$...
In particular, considering the family $\mathcal{L} =
\langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$, we give the
weight distribution of four parametrized families of cyclic codes
$\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$ and
$\mathcal{C}_{\ell,2}$ over $\mathbb{F}_q$ with zeros $\{ \alpha^{-(q^\ell+1)}
\}$, $\{ 1,\, \alpha^{-(q^\ell+1)} \}$, $\{ \alpha^{-1},\,\alpha^{-(q^\ell+1)}
\}$ and $\{ 1,\,\alpha^{-1},\,\alpha^{-(q^\ell+1)}\}$ respectively, where $q =
p^s$ with $p$ prime, $\alpha$ is a generator of $\mathbb{F}_{q^m}^*$ and
$m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient
conditions for Artin-Schreier curves $y^p-y = xR(x) + \beta x$, $p$ prime,
associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain
several maximal and minimal such curves in the case $\mathcal{L} = \langle
x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}}
\rangle$.
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Abstract