On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

Wei Qi

doi:10.3934/amc.2022015

Article Contents

2024, Volume 18, Issue 3: 661-673. Doi: 10.3934/amc.2022015

This issue Previous Article Self-orthogonal codes from equitable partitions of distance-regular graphs Next Article On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $

On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

Wei Qi^,

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

^*Corresponding author: Wei Qi
^*Corresponding author: Wei Qi

Received: January 2021

Revised: October 2021

Early access: March 2022

Published: June 2024

Abstract / Introduction Full Text(HTML) Figure(0) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 94B15; Secondary: 94B05.

Citation:

Full Text(HTML)

Table 1.

Codes	Generators	Parameters	$C^{\circ}$	Codes	Generators	Parameters	$C^{\circ}$
$C_1$	$0$	$[7,0,0] $	$C_{12}$	$C_{7}$	$g_1g_3$	$[7,3,4]$	$C_6$
$C_2$	$g_1$	$[7,6,2]$	$C_{11}$	$C_{8}$	$g_2g_3$	$[7,2,3]$	$C_5$
$C_3$	$g_2$	$[7,5,2]$	$C_{10}$	$C_{9}$	$g_1^2g_2$	$[7,3,2]$	$C_4$
$C_4$	$g_3$	$[7,4,3]$	$C_{9}$	$C_{10}$	$g_1^2g_3$	$[7,2,4]$	$C_3$
$C_5$	$g_1^2$	$[7,5,2]$	$C_{8}$	$C_{11}$	$g_1g_2g_3$	$[7,1,4]$	$C_2$
$C_6$	$g_1g_2$	$[7,4,2]$	$C_{7}$	$C_{12}$	$1$	$[7,7,1]$	$C_1$

| Show Table

DownLoad: CSV

Table 2.

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. Alahmadi, S. Dougherty, A. 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. Alahmadi, C. Güneri, H. 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-Tabue, E. 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. Gao, F. 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. Hou, S. 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-Permouth, H. Özadam, F. 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-Permouth, B. 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. Shi, H. Zhu, L. Qian, L. 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.