Combinatorics

Discrete mathematics

Aerospace engineering

Pure mathematics

Equivalence (formal languages)

Mathematics

Hull

Intersection (aeronautics)

Euclidean algorithm

Algorithm

Euclidean geometry

Linear span

Algebra over a field

Engineering

Block code

Marine engineering

Linear code

Geometry

Notation

Decoding methods

Arithmetic

IEEE Transactions on Information Theory

The Euclidean hull of a linear code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> is the intersection of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> with its Euclidean dual <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C^\perp $ </tex-math></inline-formula> . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.

On Linear Codes With One-Dimensional Euclidean Hull and Their Applications to EAQECCs