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

RICARDO A. PODESTÁ; DENIS E. VIDELA

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2021 vol. 62 p. 219 - 242

0041-6932

EN PRENSA https://inmabb.criba.edu.ar/revuma/revuma.php?p=inpressWe consider cyclic codes $CC_LL$ associated to quadratic trace forms in $m$ variables $Q_R(x) = r_{q^m/q}(xR(x))$ determined by a family $LL$ of $q$-linearized polynomials $R$ over $f_{q^m}$, and three related codes $CC_{LL,0}$, $CC_{LL,1}$ and $CC_{LL,2}$.We describe the spectra for all these codes when $LL$ is an even rank family, in terms of the distribution of ranks of the forms $Q_R$ in the family $LL$, and we also compute the complete weight enumerator for $CC_LL$. In particular, considering the family $LL = langle x^{q^ell} angle$, with $ell$ fixed in $mathbb{N}$, we give the weight distribution of four parametrized families of cyclic codes $CC_ell$, $CC_{ell,0}$, $CC_{ell,1}$ and $CC_{ell,2}$ 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 $y^p-y = xR(x) + eta x$, $p$ prime, associated to polynomials $R in LL$ to be optimal.We then obtain several maximal and minimal such curves in the case $LL = langle x^{p^ell}angle$ and $LL = langle x^{p^ell}, x^{p^{3ell}} angle$.