Mathematics Research Institute

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

Ricardo A. Podestá (Universidad Nacional de Córdoba (Argentina))

Fecha: 19/05/2020 16:00
Lugar: Webex (número de reunión 841 907 577)
Grupo: GIR SINGACOM

Abstract:

We consider cyclic codes $C_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \mbox{tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related codes $C_{\mathcal{L},0}$, $C_{\mathcal{L},1}$ and $C_{\mathcal{L},2}$. 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 $C_\mathcal{L}$. 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 $C_\ell$, $C_{\ell,0}$, $C_{\ell,1}$ and $C_{\ell,2}$ 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 $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$..

El seminario tendrá lugar en Webex:
Número de reunión: 841 907 577
https://profevirtual.webex.com/profevirtual/j.php?MTID=m0f3338222bf960938339ffa63cb27c8f

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