# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020096

## Additive and linear conjucyclic codes over ${\mathbb{F}}_4$

 1 Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE 2 University of Scranton, Scranton, PA 18510, USA

* Corresponding author: Steven T. Dougherty

Received  November 2019 Revised  February 2020 Published  July 2020

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 ${\mathbb{F}}_{4}$ 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 ${\mathbb{F}}_{4}$ and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length $n$ over ${\mathbb{F}}_{4}$ are isomorphic to binary cyclic codes of length $2n.$

Citation: Taher Abualrub, Steven T. Dougherty. Additive and linear conjucyclic codes over ${\mathbb{F}}_4$. Advances in Mathematics of Communications, doi: 10.3934/amc.2020096
##### References:
 [1] A. R. Calderbank, E. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over ${\mathbb{F}}_{4}$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.  Google Scholar [2] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar [3] S. T. Dougherty, J.-L. Kim and N. Lee, Additive self-dual codes over fields of even order, Bull. Korean Math. Soc., 55 (2018), 341-357.  doi: 10.4134/BKMS.b160842.  Google Scholar [4] S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress. Google Scholar [5] T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.  Google Scholar [6] J.-L. Kim and N. Lee, Secret sharing schemes based on additive codes over ${\mathbb{F}}_{4}$, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 79-97.  doi: 10.1007/s00200-016-0296-5.  Google Scholar [7] D. Radkova and A. J. Van Zanten, Constacyclic codes as invariant subspaces, Linear Algebra Appl., 430 (2009), 855-864.  doi: 10.1016/j.laa.2008.09.036.  Google Scholar

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##### References:
 [1] A. R. Calderbank, E. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over ${\mathbb{F}}_{4}$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.  Google Scholar [2] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar [3] S. T. Dougherty, J.-L. Kim and N. Lee, Additive self-dual codes over fields of even order, Bull. Korean Math. Soc., 55 (2018), 341-357.  doi: 10.4134/BKMS.b160842.  Google Scholar [4] S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress. Google Scholar [5] T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.  Google Scholar [6] J.-L. Kim and N. Lee, Secret sharing schemes based on additive codes over ${\mathbb{F}}_{4}$, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 79-97.  doi: 10.1007/s00200-016-0296-5.  Google Scholar [7] D. Radkova and A. J. Van Zanten, Constacyclic codes as invariant subspaces, Linear Algebra Appl., 430 (2009), 855-864.  doi: 10.1016/j.laa.2008.09.036.  Google Scholar
Conjucyclic codes of length 3
 The code $C$ Basis $V_{1}=0$ $V_{M}=\mathbb{F}_{4}^{3}$ $\left\{ \left( 1,0,0\right) ,\left( w,0,0\right) ,\left( 0,1,0\right) ,\left( 0,w,0\right) ,\left( 0,0,1\right) ,\left( 0,0,w\right) \right\}$ $V_{f}$ $\left\{ \left( 1,1,1\right) \right\}$ $V_{g}$ $\left\{ \left( 1,1,0\right) ,\left( 1,0,1\right) \right\}$ $V_{f^{2}}$ $\left\{ \left( 1,1,1\right) ,\left( w,1+w,w\right) \right\}$ $V_{g^{2}}$ $\left\{ \left( 1,1,0\right) ,\left( 1+\omega ,\omega ,0\right) ,\left( 1,0,1\right) ,\left( \omega ,0,\omega \right) \right\}$ $V_{fg}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) \right\}$ $V_{f^{2}g}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,w\right) \right\}$ $V_{fg^{2}}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,0\right) ,\left( w,0,w\right) \right\}$
 The code $C$ Basis $V_{1}=0$ $V_{M}=\mathbb{F}_{4}^{3}$ $\left\{ \left( 1,0,0\right) ,\left( w,0,0\right) ,\left( 0,1,0\right) ,\left( 0,w,0\right) ,\left( 0,0,1\right) ,\left( 0,0,w\right) \right\}$ $V_{f}$ $\left\{ \left( 1,1,1\right) \right\}$ $V_{g}$ $\left\{ \left( 1,1,0\right) ,\left( 1,0,1\right) \right\}$ $V_{f^{2}}$ $\left\{ \left( 1,1,1\right) ,\left( w,1+w,w\right) \right\}$ $V_{g^{2}}$ $\left\{ \left( 1,1,0\right) ,\left( 1+\omega ,\omega ,0\right) ,\left( 1,0,1\right) ,\left( \omega ,0,\omega \right) \right\}$ $V_{fg}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) \right\}$ $V_{f^{2}g}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,w\right) \right\}$ $V_{fg^{2}}$ $\left\{ \left( 1,0,0\right) ,\left( 0,1,0\right) ,\left( 0,0,1\right) ,\left( w,w,0\right) ,\left( w,0,w\right) \right\}$
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