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Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [ 1 ] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field F 4 F 4 and produce a duality for which the orthogonal, with respect to that duality, of conjucyclic codes is conjucyclic. Moreover, we show that this is not the case for other standard dualities. We show that additive conjucyclic codes are the only non-trivial conjucyclic codes over F 4 F 4 and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length n n over F 4 F 4 are isomorphic to binary cyclic codes of length 2 n. 2 n. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $ a $ in $ \mathbb{F}_{2}^{n}. $ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $ C $ is a right polycyclic code induced by a vector $ a\in \mathbb{F}_{2}^{n} $, then the Hermitian dual of $ C $ is a sequential code induced by $ a. $ As an application of these codes, we present examples of additive right polycyclic codes over $ \mathbb{F}_{4} $ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over $ \mathbb{F}_{4}. $ [ABSTRACT FROM AUTHOR]