Characterizations and Properties of Monic Principal Skew Codes over Rings

Let <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>A</mi> </math> be a ring with identity, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>σ</mi> </math> a ring endomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>A</mi> </math> that maps the identity to itself, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>δ</mi> </math> a <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>σ</mi> </math> -derivation of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>A</mi> </math> , and consider the skew-polynomial ring <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>A</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>X</mi> <mo>;</mo> <mi>σ</mi> <mo>,</mo> <mi>δ</mi> </mrow> </mfenced> </math> . When <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi>A</mi> </math> is a finite field, a Galois ring, or a general ring, some fairly recent literature used <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>A</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>X</mi> <mo>;</mo> <mi>σ</mi> <mo>,</mo> <mi>δ</mi> </mrow> </mfenced> </math> to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>f</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>δ</mi> </mrow> </mfenced> </math> -codes, where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>f</mi> <mo>∈</mo> <mi>A</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>X</mi> <mo>;</mo> <mi>σ</mi> <mo>,</mo> <mi>δ</mi> </mrow> </mfenced> </math> is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <mi mathvariant="script">C</mi> </math> . When <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <mi>A</mi> </math> is a finite commutative ring and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14"> <mi>σ</mi> </math> is a ring automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15"> <mi>A</mi> </math> , we also give recursive formulas for the entries of a parity-check matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16"> <mi mathvariant="script">C</mi> </math> . Also, in this case, with <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17"> <mi>δ</mi> <mo>=</mo> <mn>0</mn> </math> , we present a characterization of monic principal <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18"> <mi>σ</mi> </math> -codes whose dual codes are also monic principal <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19"> <mi>σ</mi> </math> -codes, and we deduce a characterization of self-dual monic principal <math xmlns="http://www.w3.org/1998/Math/MathML" id="M20"> <mi>σ</mi> </math> -codes. Some corollaries concerning monic principal <math xmlns="http://www.w3.org/1998/Math/MathML" id="M21"> <mi>σ</mi> </math> -constacyclic codes are also given, and a good number of highlighting examples is provided.