| Summary: | In this article, we consider a semi-local ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>=</mo><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub><mo>+</mo><mi>u</mi><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><mi>u</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><msup><mi>p</mi><mi>s</mi></msup></mrow></semantics></math></inline-formula> and <i>p</i> is a prime number. We define a multiplication <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mi>b</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>b</mi><mo>)</mo><mi>y</mi><mo>+</mo><mi>γ</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is an automorphism and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-derivation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula> so that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula>. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.
|
|---|