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Journal of Applied Mathematics and Computing

Additive conjucyclic codes over a class of Galois rings

Habibul Islam, Dipak K. Bhunia

Published Dec 21, 2023• 1 citation
Publisher

Abstract

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 $${\mathbb {F}}_{q^2}$$ Fq2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $$GR(2^r,2)$$ GR(2r,2) , where $$r\ge 2$$ r≥2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of lengthnover $$GR(2^r,2)$$ GR(2r,2) and the family of linear cyclic codes of length 2nover $${\mathbb {Z}}_{2^r}$$ Z2r . This correspondence helps to obtain additive conjucyclic codes over $$GR(2^r,2)$$ GR(2r,2) via known linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ Z2r . We prove that the trace dual $${\mathscr {C}}^{Tr}$$ CTr of an additive conjucyclic code $${\mathscr {C}}$$ C 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 $${\mathbb {Z}}_{2^r}$$ Z2r contained in additive conjucyclic codes over $$GR(2^r,2)$$ GR(2r,2) . Last but not least, we explicitly derive the generator matrices for these codes.

DOI
10.1007/s12190-023-01962-9

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