doi: 10.3934/amc.2022015
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On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255000, China

*Corresponding author: Wei Qi

Received  January 2021 Revised  October 2021 Early access March 2022

In this article, we mainly study the polycyclic codes over $ S $, where $ S = \mathbb{F}_q+u\mathbb{F}_q $ with $ u^2 = u $. First, the annihilator self-dual codes, annihilator self-orthogonal codes and annihilator $ {{{\rm{LCD}}}} $ codes over $ S $ are also introduced and studied. Next, we define a Gray map from $ S^n $ to $ \mathbb{F}^{2n}_q $ and investigate the structure properties of polycyclic codes over $ S $ using the decomposition method. The Hamming distances of the Gray images are also determined by their decompositions. Finally, we obtain some good codes based on the results.

Citation: Wei Qi. On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $. Advances in Mathematics of Communications, doi: 10.3934/amc.2022015
References:
[1]

A. AlahmadiS. DoughertyA. Leroy and P. SolÉ, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921-929.  doi: 10.3934/amc.2016049.

[2]

A. AlahmadiC. GüneriH. Shoaib and P. Solé, Long quasi-polycyclic $t$-CIS codes, Adv. Math. Commun., 12 (2018), 189-198.  doi: 10.3934/amc.2018013.

[3]

T. Blackford, The Galois variance of constacyclic codes, Finite Fields Appl., 47 (2017), 286-308.  doi: 10.1016/j.ffa.2017.06.001.

[4]

M. A. de Boer, Almost MDS codes, Desi., Codes Cryptogr., 9 (1996), 143-155.  doi: 10.1007/BF00124590.

[5]

S. T. Dougherty, Algebrais Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.

[6]

A. Fotue-TabueE. Martínez-Moro and J. T. Blackford, On polycyclic codes over a finite chain ring, Adv. Math. Commun., 14 (2020), 445-466.  doi: 10.3934/amc.2020028.

[7]

J. GaoF. Ma and F.-W. Fu, Skew constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q$, Appl. Comput. Math., 16 (2017), 286-295. 

[8]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Dev., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[9]

X.-D. HouS. R. López-Permouth and B. Parra-Avila, Rational power series sequential codes and periodicity of sequences, J. Pure Appl. Algebra, 213 (2009), 1157-1169.  doi: 10.1016/j.jpaa.2008.11.011.

[10] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003.  doi: 10.1017/CBO9780511807077.
[11]

S. R. López-PermouthH. ÖzadamF. Özbudak and S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl., 19 (2013), 16-38.  doi: 10.1016/j.ffa.2012.10.002.

[12]

S. R. López-PermouthB. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234.  doi: 10.3934/amc.2009.3.227.

[13]

M. Matsuoka, $\theta$-Polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70. 

[14]

M. Özen and H. İnce, MDS codes from polycyclic codes over finite fields, Journal of Science and Technology, 7 (2017), 55-78. 

[15]

M. Shi, X. Li, Z. Sepasdar and P. Solé, Polycyclic codes as invariant subspaces, Finite Fields Appl., 68 (2020), 101760, 14 pp. doi: 10.1016/j.ffa.2020.101760.

[16]

M. ShiH. ZhuL. QianL. Sok and P. Solé, On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q +u\mathbb{F}_q$, Cryptogr. Commun., 12 (2020), 53-70.  doi: 10.1007/s12095-019-00363-9.

show all references

References:
[1]

A. AlahmadiS. DoughertyA. Leroy and P. SolÉ, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921-929.  doi: 10.3934/amc.2016049.

[2]

A. AlahmadiC. GüneriH. Shoaib and P. Solé, Long quasi-polycyclic $t$-CIS codes, Adv. Math. Commun., 12 (2018), 189-198.  doi: 10.3934/amc.2018013.

[3]

T. Blackford, The Galois variance of constacyclic codes, Finite Fields Appl., 47 (2017), 286-308.  doi: 10.1016/j.ffa.2017.06.001.

[4]

M. A. de Boer, Almost MDS codes, Desi., Codes Cryptogr., 9 (1996), 143-155.  doi: 10.1007/BF00124590.

[5]

S. T. Dougherty, Algebrais Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.

[6]

A. Fotue-TabueE. Martínez-Moro and J. T. Blackford, On polycyclic codes over a finite chain ring, Adv. Math. Commun., 14 (2020), 445-466.  doi: 10.3934/amc.2020028.

[7]

J. GaoF. Ma and F.-W. Fu, Skew constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q$, Appl. Comput. Math., 16 (2017), 286-295. 

[8]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Dev., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[9]

X.-D. HouS. R. López-Permouth and B. Parra-Avila, Rational power series sequential codes and periodicity of sequences, J. Pure Appl. Algebra, 213 (2009), 1157-1169.  doi: 10.1016/j.jpaa.2008.11.011.

[10] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003.  doi: 10.1017/CBO9780511807077.
[11]

S. R. López-PermouthH. ÖzadamF. Özbudak and S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl., 19 (2013), 16-38.  doi: 10.1016/j.ffa.2012.10.002.

[12]

S. R. López-PermouthB. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234.  doi: 10.3934/amc.2009.3.227.

[13]

M. Matsuoka, $\theta$-Polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70. 

[14]

M. Özen and H. İnce, MDS codes from polycyclic codes over finite fields, Journal of Science and Technology, 7 (2017), 55-78. 

[15]

M. Shi, X. Li, Z. Sepasdar and P. Solé, Polycyclic codes as invariant subspaces, Finite Fields Appl., 68 (2020), 101760, 14 pp. doi: 10.1016/j.ffa.2020.101760.

[16]

M. ShiH. ZhuL. QianL. Sok and P. Solé, On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q +u\mathbb{F}_q$, Cryptogr. Commun., 12 (2020), 53-70.  doi: 10.1007/s12095-019-00363-9.

Table 1.   
Codes Generators Parameters $C^{\circ}$ Codes Generators Parameters $C^{\circ}$
$C_1$ $0$ $[7,0,0] $ $C_{12}$ $C_{7}$ $g_1g_3$ $[7,3,4]$ $C_6$
$C_2$ $g_1$ $[7,6,2]$ $C_{11}$ $C_{8}$ $g_2g_3$ $[7,2,3]$ $C_5$
$C_3$ $g_2$ $[7,5,2]$ $C_{10}$ $C_{9}$ $g_1^2g_2$ $[7,3,2]$ $C_4$
$C_4$ $g_3$ $[7,4,3]$ $C_{9}$ $C_{10}$ $g_1^2g_3$ $[7,2,4]$ $C_3$
$C_5$ $g_1^2$ $[7,5,2]$ $C_{8}$ $C_{11}$ $g_1g_2g_3$ $[7,1,4]$ $C_2$
$C_6$ $g_1g_2$ $[7,4,2]$ $C_{7}$ $C_{12}$ $1$ $[7,7,1]$ $C_1$
Codes Generators Parameters $C^{\circ}$ Codes Generators Parameters $C^{\circ}$
$C_1$ $0$ $[7,0,0] $ $C_{12}$ $C_{7}$ $g_1g_3$ $[7,3,4]$ $C_6$
$C_2$ $g_1$ $[7,6,2]$ $C_{11}$ $C_{8}$ $g_2g_3$ $[7,2,3]$ $C_5$
$C_3$ $g_2$ $[7,5,2]$ $C_{10}$ $C_{9}$ $g_1^2g_2$ $[7,3,2]$ $C_4$
$C_4$ $g_3$ $[7,4,3]$ $C_{9}$ $C_{10}$ $g_1^2g_3$ $[7,2,4]$ $C_3$
$C_5$ $g_1^2$ $[7,5,2]$ $C_{8}$ $C_{11}$ $g_1g_2g_3$ $[7,1,4]$ $C_2$
$C_6$ $g_1g_2$ $[7,4,2]$ $C_{7}$ $C_{12}$ $1$ $[7,7,1]$ $C_1$
Table 2.   
Codes generators $\Phi(C)$ $C^{\circ}$ Codes generators $\Phi(C)$ $C^{\circ}$
$C_1$ $0$ $[8,0,0] $ $C_{16}$ $C_{9}$ $Ag_2$ $[8,1,3]$ $C_8$
$C_2$ $Bf_1$ $[8,3,2]$ $C_{15}$ $C_{10}$ $Ag_2+Bf_1$ $[8,4,2]$ $C_7$
$C_3$ $Bf_2$ $[8,1,3]$ $C_{14}$ $C_{11}$ $Ag_2+Bf_2$ $[8,2,3]$ $C_6$
$C_4$ $B$ $[8,4,1]$ $C_{13}$ $C_{12}$ $Ag_2+B$ $[8,5,1]$ $C_5$
$C_5$ $Ag_1$ $[8,3,2]$ $C_{12}$ $C_{13}$ $A$ $[8,4,1]$ $C_4$
$C_6$ $Ag_1+Bf_1$ $[8,6,2]$ $C_{11}$ $C_{14}$ $A+Bf_1$ $[8,7,1]$ $C_3$
$C_7$ $Ag_1+Bf_2$ $[8,4,2]$ $C_{10}$ $C_{15}$ $A+Bf_2$ $[8,5,1]$ $C_2$
$C_8$ $Ag_1+B$ $[8,7,1]$ $C_{9}$ $C_{16}$ $A+B$ $[8,8,1]$ $C_1$
Codes generators $\Phi(C)$ $C^{\circ}$ Codes generators $\Phi(C)$ $C^{\circ}$
$C_1$ $0$ $[8,0,0] $ $C_{16}$ $C_{9}$ $Ag_2$ $[8,1,3]$ $C_8$
$C_2$ $Bf_1$ $[8,3,2]$ $C_{15}$ $C_{10}$ $Ag_2+Bf_1$ $[8,4,2]$ $C_7$
$C_3$ $Bf_2$ $[8,1,3]$ $C_{14}$ $C_{11}$ $Ag_2+Bf_2$ $[8,2,3]$ $C_6$
$C_4$ $B$ $[8,4,1]$ $C_{13}$ $C_{12}$ $Ag_2+B$ $[8,5,1]$ $C_5$
$C_5$ $Ag_1$ $[8,3,2]$ $C_{12}$ $C_{13}$ $A$ $[8,4,1]$ $C_4$
$C_6$ $Ag_1+Bf_1$ $[8,6,2]$ $C_{11}$ $C_{14}$ $A+Bf_1$ $[8,7,1]$ $C_3$
$C_7$ $Ag_1+Bf_2$ $[8,4,2]$ $C_{10}$ $C_{15}$ $A+Bf_2$ $[8,5,1]$ $C_2$
$C_8$ $Ag_1+B$ $[8,7,1]$ $C_{9}$ $C_{16}$ $A+B$ $[8,8,1]$ $C_1$
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