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(1 + u)-Constacyclic codes over Z 4 + uZ 4.

Haifeng Yu¹, Yu Wang¹, Minjia Shi²

¹Department of Mathematics and Physics, Hefei University, Hefei, China.

|August 27, 2016

Summary

This summary is machine-generated.

This study explores (1+u)-constacyclic codes over Z4+uZ4, introducing a new Gray map. This map transforms these codes into cyclic codes over Z4 and distance-invariant binary quasi-cyclic codes, yielding good binary code examples.

Keywords:

Constacyclic code Cyclic code Gray map Quasi-cyclic code

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Area of Science:

Coding Theory
Algebraic Coding
Finite Fields and Rings

Background:

Constacyclic codes are a significant subset of linear codes, often serving as the foundation for optimal linear codes.
The algebraic structure of codes over rings like Z4+uZ4 is crucial for understanding their properties and applications.
Gray maps are essential tools for translating codes over rings to codes over finite fields, facilitating analysis and construction.

Purpose of the Study:

To investigate (1+u)-constacyclic codes over the ring Z4+uZ4 for arbitrary lengths.
To introduce and utilize a novel Gray map for Z4+uZ4 codes.
To establish connections between (1+u)-constacyclic codes and well-known code families like cyclic and quasi-cyclic codes.

Main Methods:

Definition of a new Gray map mapping codes from Z4+uZ4 to Z4(4).
Application of the defined Gray map to transform (1+u)-constacyclic codes.
Combination with the classical Gray map from Z4 to F2(2) for binary code analysis.

Main Results:

The Z4 Gray image of an (1+u)-constacyclic code of length n over Z4+uZ4 is a cyclic code of length 4n over Z4.
The binary image of an (1+u)-constacyclic code of length n over Z4+uZ4 is a distance-invariant binary quasi-cyclic code of index 4 and length 8n.
Construction of examples of good binary codes derived from these (1+u)-constacyclic codes.

Conclusions:

The study successfully characterizes (1+u)-constacyclic codes over Z4+uZ4 using a novel Gray map.
This research provides a method for constructing good binary quasi-cyclic codes from codes over Z4+uZ4.
The findings offer valuable insights into the structure and applications of constacyclic codes in coding theory.