This paper introduces the simultaneous partialinverse problem for polynomials and develops its application to decoding interleaved Reed-Solomon codes and subfield-evaluation codes beyond half the minimum distance. The simultaneous partial-inverse problem has a unique solution (up to a scale factor), which can be computed by an efficient new algorithm, for which we also offer some variations. De...

We present a nonarchimedian method to construct hyperelliptic CM-curves of genus 2 over finite prime fields. Throughout the document we use the following conventions (this is only for the reference and use of the authors): d degree of the base field of the curve, i.e. C/F 2 d s number of isomorphism classes, in elliptic curve case s = h K n degree of an irreducible component of class invariants...

Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to...

Introduction: Polynomial functions over finite fields are important in computer science and electrical engineering in that they present a mathematical representation of digital circuits. The validity of circuit designs can be tested abstractly using polynomial representations of circuits instead of physical testing the circuits post-construction. Toward this end, we aim to find necessary and su...

An algebraic field extension of Q or Z/(p) may be regarded either as a structure in its own right, or as a subfield of its algebraic closure F (either Q or Z/(p)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on F , and characterize the sets of Turing degrees that are realized as such spectra. The results show a connection between enumerability in ...

A research and clinical subfield known as "human gene therapy" has grown rapidly since 1990, when the first human trials were approved in the United States. Using quantitative data, this paper describes and analyzes the research and commercial infrastructure, including academic centers, publications, intellectual property, and biotechnology firms, that has developed around the goal of discoveri...

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a generalization of B...

In this paper we discuss the question ”When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same R-place?”. We use this to show that if R contains a dense real closed subfield R′, then the spaces of R-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X) we prove that its space M(K) of R-places is metrizible if and only...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K, we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit aK-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G))...