arXiv subjects
Publications and source records attributed to Ricardo Toledano.
We present a simple method to establish the existence of asymptotically good sequences of iso-dual AG-codes. A key advantage of our approach, beyond its simplicity, is its flexibility, allowing it to be applied to a wide range of towers of function fields. As a result, we present a novel example of an asymptotically good sequence of iso-dual AG-codes over a finite field with 8 elements.
In a previous work general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this paper the genus of a family of wild towers of function fields will be considered together with a result with less restrictive sufficient conditions for a wild tower to have infinite genus.
In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the divisors $D$ and $G$ and on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.
Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\text{Con}_{F'/F}(\mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\text{Con}_{F'/F}(\mathcal{C})$ in terms of the parameters of $\mathcal{C}$, the ramification behavior of the places used to define $\mathcal{C}$ and the genus of $F$. In the case of unramified extensions of function fields we prove that $\text{Con}_{F'/F}(\mathcal{C})^\perp = \text{Con}_{F'/F}({\mathcal{C}}^\perp)$ when the degree of the extension is coprime to the characteristic of $\mathbb{F}_q$. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.
In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.
We study the asymptotic behavior of a family of algebraic geometry codes which are 4-quasi transitive linear codes. We prove that this family is asymptotically good over many prime fields using towers of algebraic function fields.
We study the asymptotic behavior of a family of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove, by using towers of algebraic function fields, that there are sequences of codes in this family attaining the Tsfasman-Vladut-Zink bound over finite fields of square cardinality. We give the exact length of these codes as well as explicit lower bounds for their parameters.
In this work, we give sufficient conditions in order to have finite ramification locus in sequences of function fields defined by different kind of Kummer extensions. These conditions can be easily implemented in a computer to generate several examples. We present some new examples of asymptotically good towers of Kummer type and we show that many known examples can be obtained from our general results.