[Paper Review] Complementary Dual Subfield Linear Codes Over Finite Fields

Kriangkrai Boonniyom, Somphong Jitman|arXiv (Cornell University)|May 22, 2016

Coding theory and cryptography5 references15 citations

TL;DR

This paper introduces and analyzes two families of linear codes over finite fields: Hermitian complementary dual codes and trace Hermitian complementary dual subfield linear codes, when the field size is a square ($q = r^2$). It establishes necessary and sufficient conditions for a code to be complementary dual under these structures, provides constructive methods for generating such codes, and presents their parameters and illustrative examples, contributing to the theory of quantum and classical error-correcting codes with improved duality properties.

ABSTRACT

Two families of complementary codes over finite fields $\mathbb{F}_q$ are studied, where $q=r^2$ is square: i) Hermitian complementary dual linear codes, and ii) trace Hermitian complementary dual subfield linear codes. Necessary and sufficient conditions for a linear code (resp., a subfield linear code) to be Hermitian complementary dual (resp., trace Hermitian complementary dual) are determined. Constructions of such codes are given together their parameters. Some illustrative examples are provided as well.

Motivation & Objective

• To investigate the structure and properties of complementary dual linear codes over finite fields where the field size is a square ($q = r^2$).
• To define and characterize two new classes: Hermitian complementary dual linear codes and trace Hermitian complementary dual subfield linear codes.
• To determine necessary and sufficient conditions for a linear code (or subfield linear code) to be Hermitian or trace Hermitian complementary dual.
• To construct explicit examples of such codes and analyze their parameters, including dimension and minimum distance.
• To contribute to the theory of self-dual and complementary dual codes with applications in quantum and classical coding theory.

Proposed method

• The paper uses the Hermitian inner product over finite fields of square order to define duality and complementarity in linear codes.
• It introduces the concept of trace Hermitian duality for subfield linear codes, extending duality beyond the full field to the base subfield.
• Necessary and sufficient conditions for a code to be Hermitian complementary dual are derived using the rank and kernel of the generator matrix under the Hermitian form.
• For subfield linear codes, the trace Hermitian dual is defined via the trace map from $\mathbb{F}_q$ to $\mathbb{F}_r$, and conditions for complementarity are established accordingly.
• Constructive methods are provided using generator matrices and algebraic constructions over $\mathbb{F}_q$, ensuring the dual code intersects trivially with the original.
• Examples are constructed explicitly using known code families and verified to satisfy the complementary dual conditions.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a linear code over $\mathbb{F}_q$ ($q = r^2$) to be Hermitian complementary dual?
• RQ2 How can the concept of duality be extended from full-field to subfield linear codes using the trace map to define trace Hermitian duality?
• RQ3 What are the structural and parameter properties of codes that are both subfield linear and trace Hermitian complementary dual?
• RQ4 Can explicit constructions of such codes be provided with known parameters (dimension, minimum distance)?
• RQ5 How do these constructions relate to existing families of self-dual or complementary dual codes in coding theory?

Key findings

• Necessary and sufficient conditions for a linear code over $\mathbb{F}_q$ ($q = r^2$) to be Hermitian complementary dual are given in terms of the rank and kernel of the generator matrix under the Hermitian inner product.
• For subfield linear codes, the paper establishes equivalent conditions for trace Hermitian complementary duality using the trace map from $\mathbb{F}_q$ to $\mathbb{F}_r$.
• Explicit constructions of Hermitian complementary dual codes are provided, including examples with specific parameters such as $[n,k,d]$ and $q = r^2$, demonstrating feasibility.
• The paper presents trace Hermitian complementary dual subfield linear codes with parameters derived from known classical codes, showing their existence and structure.
• Illustrative examples confirm the validity of the theoretical conditions and constructions, including cases where the dual code intersects trivially with the original code.
• The results extend the class of known complementary dual codes to include subfield linear codes with trace Hermitian duality, enriching the landscape of quantum and classical code design.