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• W2114905741 abstract "We study all constacyclic codes of length 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</sup> over GR <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(Rfr,m</i> ), the Galois extension ring of dimension m of the ring <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>Rfr=</i> F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> +uF <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> . The units of the ring GR <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(Rfr,m</i> ) are of the forms <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>alpha</i> , and <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>alpha+u</i> beta, where <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>alpha,</i> <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>beta</i> are nonzero elements of F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> m, which correspond to <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</i> <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>m</sup> (2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>m</sup> -1) such constacyclic codes. First, the structure and Hamming distances of <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(1+u</i> gamma)-constacyclic codes are established. We then classify all cyclic codes of length 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</sup> over <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>GR(Rfr,m</i> ), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>alpha</i> -constacyclic codes, as well as <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(1+u</i> gamma)-constacyclic and <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(alpha+u</i> beta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(1+u</i> gamma)-constacyclic accordingly to all constacyclic codes of length 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</sup> over <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>GR(Rfr,m</i> )." @default.
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• W2114905741 date "2009-04-01" @default.
• W2114905741 modified "2023-10-15" @default.
• W2114905741 title "Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${BBF}_{2}+u{BBF}_2$" @default.
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