Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes

Arezoo Soufi Karbaski; Taher Abualrub; Nuh Aydin; Peihan Liu

doi:10.3934/amc.2022004

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2024, Volume 18, Issue 1: 116-127. Doi: 10.3934/amc.2022004

This issue Previous Article Two classes of cyclic extended double-error-correcting Goppa codes Next Article Correcting adversarial errors with generalized regenerating codes

Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes

1.
Department of Mathematics, Bu Ali Sina University, Hamedan, Iran
2.
Department of Mathematics & Statistics, American University of Sharjah, Sharjah, UAE
3.
Department of Mathematics & Statistics, Kenyon College, Gambier, OH USA
4.
Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

^*Corresponding author: Nuh Aydin
^*Corresponding author: Nuh Aydin

Received: August 2021

Revised: November 2021

Early access: February 2022

Published: February 2024

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Abstract

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}. $

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Mathematics Subject Classification: Primary: 94B60, 58F17; Secondary: 94B05.

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Table 1. Additive codes $ [n, k, d]_4 $ v.s. BKLC $ [n, k, d]_4 $ with smaller dimension

$ [n, \frac{2k+1}{2}, d]_4 $	$ [n, k, d]_4 $	$ [n, k+1, d-1]_4 $	$ \langle \alpha g_1 +g_2, b\rangle $	Multinomial
$ [ 7, 9/2, 3]_4 $	$ [ 7, 4, 3 ]_4 $	$ [ 7, 5, 2 ]_4 $	$ \langle\alpha (x^2 + x + 1)+x, x^3 + x^2 + 1\rangle $	$ x^7 + x^6 + x^5 + x^3 + 1 $
$ [7, 7/2, 4]_4 $	$ [7, 3, 4]_4 $	$ [7, 4, 3]_4 $	$ \langle\alpha (x + 1)+ x^4 + x^3 + x^2 + x, x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle $	$ x^7 + 1 $
$ [22, 37/2, 3]_4 $	$ [22, 18, 3]_4 $	$ [22, 19, 2]_4 $	$ \langle\alpha (x + 1)+ x^3 + x, x^6 + x^4 + x^3 + \\x + 1\rangle $	$ x^{22} + x^{19} + x^{15} + x^{14} + x^{13} + x^8 + \\x^7 + x^6 + x^4 + x^2 + x + 1 $
$ [23, 39/2, 3]_4 $	$ [23, 19, 3]_4 $	$ [23, 20, 2]_4 $	$ \langle\alpha (x + 1)+ x^4 + x^2, x^6 + x^5 + 1\rangle $	$ x^{23} + x^{22} + x^{21} + x^{15} + x^{13} + x^{11} + \\x^{10} + x^9 + x^8 + x^7 + x^3 + x^2+ x + 1 $
$ [24, 41/2 , 3]_4 $	$ [24, 20, 3]_4 $	$ [24, 21, 2]_4 $	$ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $	$ x^{24} + x^{21} + x^{20} + x^{19} + x^{18} + x^{16} +\\ x^{14} + x^8 + x^5 + x^4 + x^3 + x^2 1, $
$ [25, 43/2, 4]_4 $	$ [25, 21, 3]_4 $	$ [25, 22, 2]_4 $	$ \langle\alpha (x + 1)+x^4 + x^2, x^6 + x^5 + 1\rangle $	$ x^{25} + x^{24} + x^{22} + x^{21} + x^{19} + x^{18} + \\x^{15} + x^{13} + x^{12} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1 $
$ [26, 45/2, 3]_4 $	$ [26, 22, 3]_4 $	$ [26, 23, 2]_4 $	$ \langle\alpha (x + 1)+x^4 + x^3 + x, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $	$ x^{26} + x^{24} + x^{21} + x^{16} + x^{15} + x^{13} + \\x^{12} + x^{11} + x^{10} + x^9 + x^7 +x^3 + \\x^2 + 1 $
$ [27, 47/2, 3]_4 $	$ [27, 23, 3]_4 $	$ [27, 24, 2]_4 $	$ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $	$ x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1 $
$ [27, 47/2, 3]_4 $	$ [27, 23, 3]_4 $	$ [27, 24, 2]_4 $	$ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $	$ x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1 $
$ [28, 49/2, 3, ]_4 $	$ [28, 24, 3]_4 $	$ [28, 25, 2]_4 $	$ \langle\alpha (x^2 + x + 1)+ x^3 + x^2 + x, x^5 + \\x^3 + x^2 + x + 1\rangle $	$ x^{28} + x^{27} + x^{24} + x^{23} + x^{22} + x^{20} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{11} + \\x^{10} + x^8 + x^7 + x^5 + x^3 + x + 1 $
$ [29, 51/2, 3]_4 $	$ [29, 25, 3]_4 $	$ [29, 26, 2]_4 $	$ \langle\alpha (x + 1)+ x^2 + x + 1, x^6 + x^5 + \\x^4 + x + 1\rangle $	$ x^{29} + x^{25} + x^{22} + x^{21} + x^{19} + x^{15} + \\x^{14} + x^{13} + x^{11} + x^{10} + x^6 +x^3 + \\x + 1 $
$ [30, 53/2, 3]_4 $	$ [30, 26, 3]_4 $	$ [30, 27, 2]_4 $	$ \langle\alpha (x + 1)+ x^3 + x^2 + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $	$ x^{30} + x^{27} + x^{26} + x^{25} + x^{24} + x^{20} + \\x^{19} + x^{18} + x^{16} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^2 + 1 $
$ [31, 55/2, 3]_4 $	$ [31, 27, 3]_4 $	$ [31, 28, 2]_4 $	$ \langle\alpha ( x + 1)+ x^4 + x + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $	$ x^{31} + x^{25} + x^{23} + x^{22} + x^{20} + x^{18} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{12} + \\x^{10} + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 +1 $

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Table 2. Optimal binary linear codes $ [n, k, d]_2 $ obtained from quaternary additive codes based on $ W $, $ T $, and $ L $

$ [n, k, d]_2 $	Map	$ \langle \alpha g_1 +g_2, b\rangle $	Multinomial
$ [ 7, 2, 4]_2^{*\circ} $	L	$ \langle\alpha (x^5 + x^4 + x + 1) + 1, x^2 + x + 1 \rangle $	$ x^7 + x^4 + x^3 + 1 $
$ [ 10, 4, 4]_2 $	L	$ \langle\alpha (x + 1)+x + 1, x^6 + x^5 + x^4 + \\x + 1\rangle $	$ x^{10} + x^9 + x^8 + x^6 + x + 1 $
$ [ 12, 5, 4]_2 $	L	$ \langle\alpha (x^2 + x + 1)+x^2 + x + 1, x^7 + \\x^3 + x^2 + x + 1\rangle $	$ x^{12} + x^6 + x^5 + x^4 + 1 $
$ [ 16, 9, 4]_2^* $	T	$ \langle\alpha (x^7 + x^3 + x^2 + x + 1)+x^4 + \\x^3 + x^2 + x + 1, x^9 + x + 1 \rangle $	$ x^{16} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + \\x^7 + x^4 + 1 $
$ [ 17, 9, 5]_2 $	T	$ \langle\alpha (x^8 + x^5 + x^4 + x^3 + 1)+x^7 + \\x^6 + x^4 + x^3 + x, x^9 + x^8 + x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle $	$ x^{17} + x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + \\x^9 + x^8 + x^7 + x^4 + x^2 + x + 1 $
$ [ 20, 11, 5]_2 $	W	$ \langle\alpha (x+1)+x^6 + x^4 + x^2 + x, x^8 + \\x^6 + x^5 + x^4 + x^2 + x + 1\rangle $	$ x^{10} + x^7 + x^5 + x^3 + x + 1 $
$ [ 26, 17, 4]_2 $	W	$ \langle\alpha (x+1)+x^4 + x^3 + x^2, x^8 + x^5 + \\x^3 + x + 1\rangle $	$ x^{13} + x^{10} + x^6 + x^3 + x + 1 $
$ [ 35, 24, 5]_2^* $	T	$ \langle\alpha (x^{11} + x^9 + x^8 + x^6 + x^4 + x^3 + \\x^2 + x + 1)+x^{22} + x^{16} + x^{15} + x^{14} + \\x^{13} + x^{10} + x^9 + x^8 + x^7 + x^5 + x + \\1 , x^{24} + x^{21} + x^{19} + x^{16} + x^{13} + \\x^{12} + x^{10} + x^8 + x^7 + x^6 + x^2 + x +1\rangle $	$ x^{35} + x^{33} + x^{30} + x^{29} + x^{26} + x^{23} + \\x^{21} + x^{20} + x^{19} + x^{14} + x^{13} +x^{12} + \\x^{11} + x^9 + x^8 + x^7 + x^6 + x^4 + \\x^3 + x^2 + 1 $
$ [ 49, 39, 4]_2^* $	T	$ \langle\alpha ( x^{10} + x^8 + x^6 + x^4 + x^2 + x + \\1)+x , x^5 + x^4 + x^3 + x^2 + 1 \rangle $	$ x^{49} + x^{46} + x^{45} + x^{44} + x^{40} + x^{39} + \\x^{38} + x^{36} + x^{35} + x^{31} + x^{30} + x^{29} + \\x^{28} + x^{27} + x^{24} + x^{23} + x^{20} + x^{19} + \\x^{17} + x^{15} + x^{12} + x^{11} + x^9 + x^8 + \\x^4 + x + 1 $
$ [ 62, 51, 4]_2 $	W	$ \langle\alpha ( x + 1)+x^8 + x^4 + x + 1 , x^10 + \\x^9 + x^8 + x^4 + 1 \rangle $	$ x^{31} + x^{29} + x^{28} + x^{27} + x^{26} + x^{25} + \\x^{23} + x^{15} + x^{14} + x^{13} + x^{12} +x^9 + \\x^8 + x^7 + x^3 + x^2 + x + 1 $
$ [ 98, 86, 4]_2 $	W	$ \langle\alpha (x^2+ x + 1)+x^8 + x^7 + x^6 + x^3 + \\1 , x^{10} + x^8 + x^6 + x^4 + x^2 + x + 1\rangle $	$ x^{49} + x^{48} + x^{46} + x^{45} + x^{43} + x^{39} + \\x^{36} + x^{31} + x^{27} + x^{25} + x^{24} +x^{22} + \\x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{14} + \\x^7 + x^6 + x^3 + x + 1 $

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Table 3. Optimal quantum codes $ [[n, k, d]]_4 $ from self-dual/self-orthogonal/dual-containing binary linear codes obtained from quaternary additive codes

$ [[n, k, d]]_4 $	Map	$ \langle \alpha g_1 +g_2, b\rangle $	Multinomial
$ [[ 7, 1, 3]]_4 $	T	$ \langle\alpha (x^3+x^2 + 1)+ x, x^4 + x^3 + x^2 + x + 1 \rangle $	$ x^7 + x^4 + x^3 + x + 1 $
$ [[ 10, 8, 2]]_4 $	T	$ \langle\alpha (x + 1)+ 1, x^2 + x + 1\rangle $	$ x^{10} + x^9 + x^7 + x^5 + x^4 + x^2 + x + 1 $
$ [[ 15, 7, 3]]_4 $	T	$ \langle\alpha (x^4 + x + 1)+ x^2 + x + 1, x^4 + x^3 + 1 \rangle $	$ x^{15} + x^{10} + x^9 + x^7 + x^6 + x^5 + \\x^3 + x^2 + 1 $
$ [[ 30, 26, 2]]_4 $	T	$ \langle\alpha (x^2 + x + 1)+ x^8 + x^6 + 1, x^{12} + x^{11} + \\x^8 + x^7 + x^5 + x^4 + x^2 + x + 1 \rangle $	$ x^{30} + x^{29} + x^{27} + x^{25} + x^{24} + x^{23} + \\x^{22} + x^{21} + x^{20} + x^{17} + x^{13} + x^{11} + \\x^{10} + x^9 + x^6 + x^5 + x^2 + x + 1 $
$ [[ 35, 29, 2]]_4 $	L	$ \langle\alpha ( x + 1)+ x + 1, x^3 + x + 1 \rangle $	$ x^{35} + x^{34} + x^{33} + x^{31} + x^{27} + x^{25} + \\x^{24} + x^{22} + x^{20} + x^{18} + x^{17} + x^{16} + \\x^{15} + x^{11} + x^5 + x^4 + x + 1 $
$ [[ 36, 30, 2]]_4 $	W	$ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $	$ x^{18} + x^{16} + x^{13} + x^{11} + x^9 + x^8 + \\x^7 + x^4 + x^3 + 1 $
$ [[ 48, 42, 2]]_4 $	W	$ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $	$ x^{24} + x^{22} + x^{16} + x^{15} + x^{14} + x^{13} + \\x^{11} + x^{10} + x^7 + x^6 + x^4 + 1 $
$ [[ 56, 50, 2]]_4 $	L	$ \langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle $	$ x^{56} + x^{51} + x^{49} + x^{48} + x^{47} + x^{41} + \\x^{38} + x^{36} + x^{31} + x^{30} + x^{29} + x^{28} + \\x^{26} + x^{18} + x^{16} + x^{15} + x^{14} + x^{12} + \\x^{10} + x^9 + x^5 + x^2 + x + 1 $
$ [[ 72, 66, 2]]_4 $	W	$ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $	$ x^{36} + x^{35} + x^{32} + x^{31} + x^{30} + x^{29} + \\x^{26} + x^{22} + x^{21} + x^{20} + x^{18} +x^{16} + \\x^{13} + x^{10} + x^9 + x^2 + x + 1 $
$ [[ 84, 76, 2]]_4 $	W	$ \langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle $	$ x^{42} + x^{41} + x^{40} + x^{39} + x^{37} + x^{35} + \\x^{33} + x^{30} + x^{29} + x^{28} + x^{27} +x^{25} + \\x^{21} + x^{15} + x^{14} + x^{13} + x^{11} + x^{10} + \\x^9 + x^6 + x^5 + x^2 + x + 1 $

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Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes

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