Weight distribution of cyclic codes defined by quadratic forms and related curves
RA Podestá, DE Videla - arXiv preprint arXiv:1903.01838, 2019 - arxiv.org
arXiv preprint arXiv:1903.01838, 2019•arxiv.org
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 $ 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 …
We consider cyclic codes associated to quadratic trace forms in variables determined by a family of -linearized polynomials over , and three related codes , and . We describe the spectra for all these codes when is an even rank family, in terms of the distribution of ranks of the forms in the family , and we also compute the complete weight enumerator for . In particular, considering the family , with fixed in , we give the weight distribution of four parametrized families of cyclic codes , , and over with zeros , , and respectively, where with prime, is a generator of and is even. Finally, we give simple necessary and sufficient conditions for Artin-Schreier curves , prime, associated to polynomials to be optimal. We then obtain several maximal and minimal such curves in the case and .
arxiv.org
以上显示的是最相近的搜索结果。 查看全部搜索结果