Additive conjucyclic codes over a class of Galois rings

Abstract As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except $${\mathbb {F}}_{q^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> </mml:math> ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $$GR(2^r,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$r\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over $$GR(2^r,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and the family of linear cyclic codes of length 2 n over $${\mathbb {Z}}_{2^r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> </mml:msub> </mml:math> . This correspondence helps to obtain additive conjucyclic codes over $$GR(2^r,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> via known linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> </mml:msub> </mml:math> . We prove that the trace dual $${\mathscr {C}}^{Tr}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>Tr</mml:mi> </mml:mrow> </mml:msup> </mml:math> of an additive conjucyclic code $${\mathscr {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> </mml:msub> </mml:math> contained in additive conjucyclic codes over $$GR(2^r,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Last but not least, we explicitly derive the generator matrices for these codes.