Skew Cyclic Codes Over ℤ₄ + <i>u</i>ℤ₄ With Derivation Based on a New Gray Map and Automorphism

This paper investigates the algebraic structure and properties of skew cyclic codes over the finite chain ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> = Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i>Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub>, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0. A central contribution of this work is the introduction and application of a novel Gray map, establishing a distance-preserving link between codes over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> and linear codes over Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub>.We employ a specific, compatible pair consisting of a ring automorphism θ and a θ-derivation η to define the appropriate skew polynomial ring structure <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i>[<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i>; θ, η]. Within this algebraic framework, we provide a comprehensive analysis of the fundamental structure of free (θ, η)-cyclic codes, detailing their generator polynomial structure and establishing their precise relationship with classical cyclic or quasi-cyclic codes. Furthermore, the structure of Euclidean dual codes for these free codes is examined for even lengths, and a construction for double (θ, η)-cyclic codes from free constituent codes is also presented.