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February  2022, 16(1): 1-15. doi: 10.3934/amc.2020096

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

Taher Abualrub 1, and  Steven T. Dougherty 2,,

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  February 2022 Early access  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. $

Keywords: Conjucyclic code, additive code, codes over $ {\mathbb F}_4 $.
Mathematics Subject Classification: Primary:11T71, 94B15.
Citation: Taher Abualrub, Steven T. Dougherty. Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $. Advances in Mathematics of Communications, 2022, 16 (1) : 1-15. 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.

[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.

[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.

[4]

S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress.

[5]

T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.

[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.

[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.

show all references

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.

[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.

[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.

[4]

S. T. Dougherty and S. Meyers, Orthogonality from Group Characters, work in progress.

[5]

T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.

[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.

[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.

Table 1.  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\} $
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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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