Weight distribution of cyclic codes defined by quadratic forms and related curves

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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