Asymptotically Bad Towers of Function Fields

Asymptotics for the genus and the number of rational places in towers of function fields over a finite field

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345---1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99---107], and Garcia [Comm. Algebra, 20(12): 3683---3689]. As an application, we show that the minimum distance corresponds to the Feng---Rao bound.

In this paper, we investigate a special tower of function fields, which is asymptotically bad. The primary goal is to construct explicitly a class of algebraic-geometry (AG) codes based on this tower. The so called Hermitian codes are codes corresponding to the 2nd function field in this tower.

In this paper, a convenient set of functions is identified whose span includes all functions in a tower of function fields of Garcia and Stichtenoth that have poles only at the place at infinity. The latter set is of interest in the construction of long and efficient “one-point” AG codes.

Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields.

The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity. Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ in terms of genus ${g_F}$ for a function field $F$ over a finite field $\mathbb{F}_q$. Although the whole group $\mbox{Aut}(F/\mathbb{F}_q)$ could have size $\Omega({g_F}^4)$, the maximum size $m_F$ of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ is upper bounded by $4g_F+4$ for $g_F\ge 2$. In the present paper, we study the asymptotic behavior of $m_F$ by defining $M_q=\limsup_{{g_F}\rightarrow\infty}\frac{m_F \cdot \log_q m_F}{{g_F}}$, where $F$ runs through all function fields over $\mathbb{F}_q$. We show that $M_q$ lies between $2$ and $3$ (or $4$) for odd characteristic (or for even characteristic, respectively). This means that $m_F$ grows much more slowly than genus does asymptotically. The second part of this paper is to study the maximum size $b_F$ of subgroups of $\mbox{Aut}(F/\mathbb{F}_q)$ whose order is coprime to $q$. The Hurwitz bound gives an upper bound $b_F\le 84(g_F-1)$ for every function field $F/\mathbb{F}_q$ of genus $g_F\ge 2$. We investigate the asymptotic behavior of $b_F$ by defining ${B_q}=\limsup_{{g_F}\rightarrow\infty}\frac{b_F}{{g_F}}$, where $F$ runs through all function fields over $\mathbb{F}_q$. Although the Hurwitz bound shows ${B_q}\le 84$, there are no lower bounds on $B_q$ in literature. One does not even know if ${B_q}=0$. For the first time, we show that ${B_q}\ge 2/3$ by explicitly constructing some towers of function fields in this paper.

On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields

Chudnovsky-type algorithms for the multiplication in finite extensions of finite fields are well-known for having a good bilinear complexity, both asymptotically and at finite distance. More precisely, for every degree n of the extension, the existence of a family of algorithms with linear bilinear complexity in n has been proved using the original method applied to an explicit recursive tower of function fields. However, there is currently no method to build these algorithms in polynomial time. Nevertheless, one can construct in polynomial time a Chudnovsky-type algorithm over the projective line for the multiplication in any extension degree, with a quasi-linear bilinear complexity. In this paper, we prove that we can obtain algorithms both constructible in polynomial time and having a linear bilinear complexity by mixing up these two strategies.

Recently, a new explicit tower of function fields was introduced by Bassa, Beelen, Garcia and Stichtenoth (BBGS). This resulted in currently the best known lower bound for Ihara’s constant in the case of non-prime finite fields. In particular over cubic fields, the tower’s limit is at least as good as Zink’s bound; i.e. λ ( BBGS / F q 3 ) ≥ 2 ( q 2 − 1 ) / ( q + 2 ) \lambda (\text {BBGS}/\mathbb {F}_{q^3})\geq 2(q^2-1)/(q+2) . In this paper, the exact value of λ ( BBGS / F q 3 ) \lambda (\text {BBGS}/\mathbb {F}_{q^3}) is computed. We also settle a question stated in Ihara’s 2007 paper.

We study the asymptotic behaviour of the genus in some Artin-Schreier towers of function fields over a finite field, and we present a new class of Artin-Schreier towers having finite genus.

The Galois closure of Drinfeld modular towers

We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Ap\'{e}ry sets, and thus several results concerning Ap\'ery sets of Arf semigroups are presented.

Low increasing tower of algebraic function fields and bilinear complexity of multiplication in any extension of [formula omitted

Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X, D) is a universal quotient of this Chow group. As an application we can rephrase Lang's class field theory of function fields of varieties over finite fields in explicit terms.