Weight distribution of cyclic codes defined by quadratic forms and related curves
نویسندگان
چکیده
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 $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ $\mathcal{C}_{\mathcal{L},2}$. describe the spectra for all these when is an even rank family, terms distribution ranks $Q_R$ $\mathcal{L}$, we also compute complete weight enumerator $\mathcal{C}_\mathcal{L}$. In particular, considering $\mathcal{L} \langle x^{q^\ell} \rangle$, with $\ell$ fixed $\mathbb{N}$, give four parametrized families $\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$ $\mathcal{C}_{\ell,2}$ $\mathbb{F}_q$ zeros $\{ \alpha^{-(q^\ell+1)} \}$, 1,\, \alpha^{-1},\,\alpha^{-(q^\ell+1)} \}$ 1,\,\alpha^{-1},\,\alpha^{-(q^\ell+1)}\}$ respectively, where $q p^s$ $p$ prime, $\alpha$ generator $\mathbb{F}_{q^m}^*$ $m/(m,\ell)$ even. Finally, simple necessary sufficient conditions Artin-Schreier curves $y^p-y xR(x) + \beta x$, $R \in \mathcal{L}$ be optimal. then obtain several maximal minimal such case x^{p^\ell}\rangle$ x^{p^\ell}, x^{p^{3\ell}} \rangle$.
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ژورنال
عنوان ژورنال: Revista De La Union Matematica Argentina
سال: 2021
ISSN: ['0041-6932', '1669-9637']
DOI: https://doi.org/10.33044/revuma.1840