Cyclotomic Function Fields Research Articles

Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic

Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic

Class fields, Dirichlet characters, and extended genus fields of global function fields

AbstractWe study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.

Genus Field and Extended Genus Field of an Elementary Abelian Extension of Global Fields

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian l-extension of a field of rational functions, where l is a prime number. In the Kummer case, if K is contained in a cyclotomic function field, the construction is given using Leopoldt’s ideas by means of Dirichlet characters. Following the definition of Anglès and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian l-extension of a field of rational functions.

Binary Sequences With a Low Correlation via Cyclotomic Function Fields

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula> for a positive integer <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. It is often overlooked in this community that all <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula> rational places (including &#x201C;the place at infinity&#x201D;) of the rational function field over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula> form a cyclic structure under an automorphism of order <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula>. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula> via cyclotomic function fields over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula>. Each family of our sequences has size <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula> and its correlation is upper bounded by <inline-formula> <tex-math notation="LaTeX">$\lfloor 2^{(n+2)/2}\rfloor $ </tex-math></inline-formula>. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula>. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula> for even <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.

Genus fields of Kummer ℓn-cyclic extensions

We give a construction of the genus field for Kummer [Formula: see text]-cyclic extensions of rational congruence function fields, where [Formula: see text] is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer [Formula: see text]-cyclic extension. Finally, we study the extension [Formula: see text], for [Formula: see text], [Formula: see text] abelian extensions of [Formula: see text].

Infinite families of cyclotomic function fields with any prescribed class group rank

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F q ( T ) whose class groups can have arbitrarily large ℓ n -rank, where F q is the finite field of prime power order q . We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k ( Λ ) + of cyclotomic function fields k ( Λ ) whose ideal class groups have arbitrary ℓ n -rank for n = 1, 2, and 3, where ℓ is a prime divisor of q − 1 . We also obtain a tower of cyclotomic function fields K i whose maximal real subfields have ideal class groups of ℓ n -ranks getting increased as the number of the finite places of k which are ramified in K i get increased for i ≥ 1 . Our main idea is to use the Kummer extensions over k which are subfields of k ( Λ ) + , where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k ( Λ ) + of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k . We demonstrate our results by presenting some examples calculated by MAGMA at the end.

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.

Representation of units in cyclotomic function fields

In this paper, we prove a refinement of Hilbert’s Satz 90 for the [Formula: see text]-cyclotomic function fields, where [Formula: see text] is a monic prime in [Formula: see text]. Our result can be viewed as a function field analogue of Newman’s theorem which is a refinement of Hilbert’s Satz 90 for [Formula: see text].

Irreducible polynomials over [formula omitted] with three prescribed coefficients

For any positive integers n≥3 and r≥1, we prove that the number of monic irreducible polynomials of degree n over F2r in which the coefficients of Tn−1, Tn−2 and Tn−3 are prescribed has period 24 as a function of n, after a suitable normalization. A similar result holds over F5r, with the period being 60. We also show that this is a phenomena unique to characteristics 2 and 5. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz.

Infinite families of irregular primes in cyclotomic function fields

We find both a lower bound and an upper bound on the p-rank of the divisor class group of the fth cyclotomic function field k(Λf) and the Jacobian of k(Λf)F¯q, where f is an irreducible polynomial in the rational function field k=Fq(t) and Fq is the finite field of order q with characteristic p. Moreover, we find two types of infinite families of irregular primes f for which the divisor class numbers of the maximal real cyclotomic function fields k(Λf)+ with conductor f are divisible by N. For the first family of irregular primes, N is equal to pp(p−1), a power of a prime, and for the second family of irregular primes, N is a composite number (pℓ)5 for a prime ℓ different from a prime p. Furthermore, in the former case, the divisor class group of k(Λf)+ has p-rank at least p(p−1).

Subspace designs based on algebraic function fields

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically $q \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over $\F^n$ for any field $\F$, with logarithmic degree and expansion guarantee for subspaces of dimension $\Omega(n/(\log \log n))$.

Indivisibility of divisor class numbers of Kummer extensions over the rational function field

We find a complete criterion for a Kummer extension K over the rational function field k=Fq(T) of degree ℓ to have indivisibility of its divisor class number hK by ℓ, where Fq is the finite field of order q and ℓ is a prime divisor of q−1. More importantly, when hK is not divisible by ℓ, we have hK≡1(modℓ). In fact, the indivisibility of hK by ℓ depends on the number of finite primes ramified in K/k and whether or not the infinite prime of k is unramified in K. Using this criterion, we explicitly construct an infinite family of the maximal real cyclotomic function fields whose divisor class numbers are divisible by ℓ.

Construction of Sequences With High Nonlinear Complexity From Function Fields

Complexity of sequences plays an important role in pseudorandom sequences and cryptography. In this paper, we present a construction of sequences with high nonlinear complexity from function fields. The main idea is to make use of function fields with many rational places as well as an automorphism of large order. We illustrate our construction through rational function fields and cyclotomic function fields in which there exist some automorphisms of large order. It turns out that we are able to: 1) slightly increase the length of the inversive sequence without losing nonlinear complexity and 2) obtain sequences with much larger nonlinear complexity than random sequences.

Regulators of an Infinite Family of the Simplest Quartic Function Fields

AbstractWe explicitly find regulators of an infinite family {Lm} of the simplest quartic function fields with a parameter m in a polynomial ring [t], where is the finite field of order q with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family {Lm} and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain {Lm} as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field (t) in {Lm}.

Explicit Galois representations of automorphisms on holomorphic differentials in characteristic p

We determine the representation of the Galois group for the cyclotomic function fields in characteristic p>0 induced by the natural action on the space of holomorphic differentials via construction of an explicit canonical basis of differentials. This includes those cases which present wild ramification and finite automorphism groups with non-cyclic p-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes and extend these results to rank one Drinfel'd modules.

Unit indices of function fields of degree four

The aim of this paper is to investigate unit indices for subfields of cyclotomic function fields. In particular, we determine unit indices of certain imaginary abelian function fields of degree four.

On automorphism groups of cyclotomic function fields over finite fields

The Weierstrass semigroup of the unique totally ramified place in the cyclotomic function field with modulus xn+1 over the rational function field Fq(x) is explicitly computed for each positive integer n. As a consequence, the automorphism groups of cyclotomic function fields with modulus xn+1 over finite fields can be determined. Similarly, the automorphism groups of the cyclotomic function fields with modulus P where P is an irreducible quadratic polynomial over finite fields are investigated as well.

Generators of finite fields with powers of trace zero and cyclotomic function fields

In this paper, we count the number of generators of finite fields with powers of trace zero up to some point, answering a question of Z. Reichstein. More generally, we count irreducible polynomials over finite fields with some prescribed coefficients. This is done by relating this count to the problem of counting rational points on curves over finite fields whose function fields are subfields of cyclotomic function fields.

Division algebras and maximal orders for given invariants

Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for$\mathbb{F}_{q}(t)$, excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

Optimal rate algebraic list decoding using narrow ray class fields

We use class field theory, specifically Drinfeld modules of rank 1, to construct a family of asymptotically good algebraic–geometric (AG) codes over fixed alphabets. Over a field of size ℓ2, these codes are within 2/(ℓ−1) of the Singleton bound. The function fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are “folded” using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev Density Theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching 1−R where R is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field.Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound — specifically, for any desired rate R∈(0,1) and constant ε>0, we get codes over an alphabet size (1/ε)O(1/ε2) that can be list decoded up to error fraction 1−R−ε confining close-by messages to a subspace with NO(1/ε2) elements. Previous results for list decoding up to error-fraction 1−R−ε over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic–geometric codes. In contrast, our result shows that these folded algebraic–geometric codes themselves have the claimed list decoding property.