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 $Q_R(x)={\mathrm{Tr}}_{q^m\mathrm{/}q}(xR(x))$ 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 $$\mathcal{L}$ = \langle x^{q^\ell} \rangle$,with$\ell$fixedin$$\mathbb{N}$$, we give the weight distribution of four parametrized families of cyclic codes ${\mathcal{C}}_{\mathrm{\ell }}$, ${\mathcal{C}}_{\mathrm{\ell },0}$, ${\mathcal{C}}_{\mathrm{\ell },1}$ and ${\mathcal{C}}_{\mathrm{\ell },2}$ over ${\mathbb{F}}_q$ with zeros ${ \alpha^{-(q^\ell+1)} }$,${ 1,, \alpha^{-(q^\ell+1)} }$,${ \alpha^{-1},,\alpha^{-(q^\ell+1)} }$and${ 1,,\alpha^{-1},,\alpha^{-(q^\ell+1)}}$respectively,where$q = p^s$with$p$prime,$\alpha$isageneratorof$$\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$.