On cyclic and negacyclic codes with one-dimensional hulls and their applications

Steven T. Dougherty; Serap Şahinkaya

doi:10.3934/amc.2022096

Article Contents

2024, Volume 18, Issue 5: 1364-1378. Doi: 10.3934/amc.2022096

This issue Previous Article RLWE/PLWE equivalence for the maximal totally real subextension of the $ 2^rpq $-th cyclotomic field Next Article Optimal wide-gap-zone frequency hopping sequences

On cyclic and negacyclic codes with one-dimensional hulls and their applications

Steven T. Dougherty^1, , and
Serap Şahinkaya^{1, 2,}

1.
Department of Mathematics, University of Scranton, Scranton, PA, 18510, USA
2.
Tarsus University, Faculty of Engineering, Department of Natural and Mathematical Sciences, Mersin, Turkey

^*Corresponding author: Steven T. Dougherty
^*Corresponding author: Steven T. Dougherty

Received: September 2022

Revised: November 2022

Early access: December 2022

Published: October 2024

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Abstract

Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.

Keywords:

Cyclic code,
nega-cyclic code,
hull.

Mathematics Subject Classification: Primary: 11T71, 94B15.

Citation:

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Table 1. Some New EAQECCs over $ {\mathbb{F}}_{2^3} $ from Cyclic Codes with one-dimensional Hulls

Structure	Generator polynomial of cyclic code	Parameters of cyclic code	Optimal/Near Optimal	New EAQECCs	EA-QMDS
$ Q_1 $	$ x + 1 $	$ [6, 5, 2]_8 $	Optimal	$ [[ 6, 4, 2;0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [6, 5, 2]_8 $	Optimal	$ [[ 6, 0, 6; 4]]_8 $	Yes
$ Q_1 $	$ x + w $	$ [7, 6, 2]_8 $	Optimal	$ [[ 7, 5, 2; 0]]_8 $	Yes
$ Q_2 $	$ x + w $	$ [7, 6, 2]_8 $	Optimal	$ [[ 7, 0, 7; 5]]_8 $	Yes
$ Q_1 $	$ x^2 + w^3x + w $	$ [7, 5, 3]_8 $	Optimal	$ [[ 7, 4, 3; 1]]_8 $	Yes
$ Q_2 $	$ x^2 + w^3x + w $	$ [7, 5, 3]_8 $	Optimal	$ [[ 7, 1, 6; 4]]_8 $	Yes
$ Q_1 $	$ x^3 + x^2 + w^5x + w $	$ [7, 4, 4]_8 $	Optimal	$ [[ 7, 3, 4; 2]]_8 $	Yes
$ Q_2 $	$ x^3 + x^2 + w^5x + w $	$ [7, 4, 4]_8 $	Optimal	$ [[ 7, 2, 5; 3]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [8, 7, 2]_8 $	Optimal	$ [[ 8, 6, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [8, 7, 2]_8 $	Optimal	$ [[ 8, 0, 8, 6]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [10, 9, 2]_8 $	Optimal	$ [[ 10, 8, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [10, 9, 2]_8 $	Optimal	$ [[ 10, 0, 10, 8]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [12, 11, 2]_8 $	Optimal	$ [[ 12, 10, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [12, 11, 2]_8 $	Optimal	$ [[ 12, 0, 12, 10]]_8 $	Yes
$ Q_1 $	$ x^3 + x^2 + x + 1 $	$ [12, 9, 2]_8 $	Near Optimal	$ [[ 12, 8, 2, 2]]_8 $	No
$ Q_2 $	$ x^3 + x^2 + x + 1 $	$ [12, 9, 2]_8 $	Near Optimal	$ [[ 12, 2, 6, 8]]_8 $	No
$ Q_1 $	$ x + w^2 $	$ [14, 13, 2]_8 $	Optimal	$ [[ 14, 12, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + w^2 $	$ [14, 13, 2]_8 $	Optimal	$ [[ 14, 0, 14, 12]]_8 $	Yes
$ Q_1 $	$ x^3 + w^6x^2 + x + w^6 $	$ [14, 11, 3]_8 $	Optimal	$ [[ 14, 10, 3, 2]]_8 $	No
$ Q_2 $	$ x^3 + w^6x^2 + x + w^6 $	$ [14, 11, 3]_8 $	Optimal	$ [[ 14, 2, 7, 10]]_8 $	No
$ Q_1 $	$ x + 1 $	$ [16, 15, 2]_8 $	Optimal	$ [[ 16, 14, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [16, 15, 2]_8 $	Optimal	$ [[ 16, 0, 16, 14]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [18, 17, 2]_8 $	Optimal	$ [[ 18, 16, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [18, 17, 2]_8 $	Optimal	$ [[ 18, 0, 18, 16]]_8 $	Yes
$ Q_1 $	$ x^5 + x^4 + w^4x^3 + w^4x^2 + x + 1 $	$ [18, 13, 4]_8 $	Near Optimal	$ [[ 18, 12, 4, 4]]_8 $	No
$ Q_2 $	$ x^5 + x^4 + w^4x^3 + w^4x^2 + x + 1 $	$ [18, 13, 4]_8 $	Near Optimal	$ [[ 18, 4, 8, 12]]_8 $	No
$ Q_1 $	$ x + 1 $	$ [20, 19, 2]_8 $	Optimal	$ [[ 20, 18, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [20, 19, 2]_8 $	Optimal	$ [[ 20, 0, 20, 18]]_8 $	Yes
$ Q_1 $	$ x^3 + x^2 + x + 1 $	$ [20, 17, 2]_8 $	Near Optimal	$ [[ 20, 16, 2, 24]]_8 $	No
$ Q_2 $	$ x^3 + x^2 + x + 1 $	$ [20, 17, 2]_8 $	Near Optimal	$ [[ 20, 2, 10, 16]]_8 $	No
$ Q_1 $	$ x^2 + wx + w^3 $	$ [21, 19, 2]_8 $	Optimal	$ [[ 21, 18, 2, 1]]_8 $	No
$ Q_2 $	$ x^2 + wx + w^3 $	$ [21, 19, 2]_8 $	Optimal	$ [[ 21, 1, 18, 18]]_8 $	No
$ Q_1 $	$ x^3 + w^6x^2 + w^6x + w^2 $	$ [21, 18, 3]_8 $	Optimal	$ [[ 21, 17, 3, 2]]_8 $	No
$ Q_2 $	$ x^3 + w^6x^2 + w^6x + w^2 $	$ [21, 18, 3]_8 $	Optimal	$ [[ 21, 2, 14, 17]]_8 $	No
$ Q_1 $	$ x^5 + w^2x^4 + w^2x^3 + w^3x^2 + w^4x + w^4 $	$ [21, 16, 4]_8 $	Optimal	$ [[ 21, 15, 4, 4]]_8 $	No
$ Q_2 $	$ x^5 + w^2x^4 + w^2x^3 + w^3x^2 + w^4x + w^4 $	$ [21, 16, 4]_8 $	Optimal	$ [[ 21, 4, 12, 15]]_8 $	No
$ Q_1 $	$ x + 1 $	$ [22, 21, 2]_8 $	Optimal	$ [[ 22, 20, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [22, 21, 2]_8 $	Optimal	$ [[ 22, 0, 22, 20]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [24, 23, 2]_8 $	Optimal	$ [[ 24, 22, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [24, 23, 2]_8 $	Optimal	$ [[ 24, 0, 24, 22]]_8 $	Yes
$ Q_1 $	$ x + 1 $	$ [26, 25, 2]_8 $	Optimal	$ [[ 26, 24, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [26, 25, 2]_8 $	Optimal	$ [[ 26, 0, 26, 24]]_8 $	Yes
$ Q_1 $	$ x + w $	$ [28, 27, 2]_8 $	Optimal	$ [[ 28, 26, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + w $	$ [28, 27, 2]_8 $	Optimal	$ [[ 28, 0, 28, 26]]_8 $	Yes
$ Q_1 $	$ x^3 + x^2 + x + 1 $	$ [28, 25, 2]_8 $	Near Optimal	$ [[ 28, 24, 2, 2]]_8 $	No
$ Q_2 $	$ x^3 + x^2 + x + 1 $	$ [28, 25, 2]_8 $	Near Optimal	$ [[ 28, 2, 14, 24]]_8 $	No
$ Q_1 $	$ x + 1 $	$ [30, 29, 2]_8 $	Optimal	$ [[ 30, 28, 2, 0]]_8 $	Yes
$ Q_2 $	$ x + 1 $	$ [30, 29, 2]_8 $	Optimal	$ [[ 30, 0, 30, 28]]_8 $	Yes

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Table 2. Some New EAQECCs over $ {\mathbb{F}}_{3^2} $ from Cyclic Codes with one-dimensional Hulls

Structure	Generator polynomial of cyclic code	Parameters of cyclic code	Optimal/Near Optimal	New EAQECCs	EA-QMDS
$ Q_1 $	$ x + w^2 $	$ [8, 7, 2]_9 $	Optimal	$ [[ 8, 6, 2;0]]_9 $	Yes
$ Q_2 $	$ x + w^2 $	$ [8, 7, 2]_9 $	Optimal	$ [[ 8, 0, 8; 6]]_9 $	Yes
$ Q_1 $	$ x^2 + w^2x + w $	$ [8, 6, 3]_9 $	Optimal	$ [[ 8, 5, 3;1]]_9 $	Yes
$ Q_2 $	$ x^2 + w^2x + w $	$ [8, 6, 3]_9 $	Optimal	$ [[ 8, 1, 7; 5]]_9 $	Yes
$ Q_1 $	$ x^4 + 2x^3 + 2x^2 + w^7x + w^6 $	$ [8, 4, 5]_9 $	Optimal	$ [[ 8, 3, 5;3]]_9 $	Yes
$ Q_1 $	$ x^3 + 2x + w^2 $	$ [8, 5, 3]_9 $	Near Optimal	$ [[ 8, 4, 3; 2]]_9 $	Yes
$ Q_2 $	$ x^3 + 2x + w^2 $	$ [8, 5, 3]_9 $	Near Optimal	$ [[ 8, 2, 5; 4]]_9 $	Yes
$ Q_1 $	$ x+2 $	$ [9, 8, 2]_9 $	Optimal	$ [[ 9, 7, 2; 0]]_9 $	Yes
$ Q_2 $	$ x+2 $	$ [9, 8, 2]_9 $	Optimal	$ [[ 9, 0, 9; 7]]_9 $	Yes
$ Q_1 $	$ x + w^2 $	$ [12, 11, 2]_9 $	Optimal	$ [[ 12, 10, 2; 0]]_9 $	Yes
$ Q_2 $	$ x + w^2 $	$ [12, 11, 2]_9 $	Optimal	$ [[ 12, 0, 2; 10]]_9 $	No
$ Q_1 $	$ x^2 + x + 1 $	$ [12, 10, 2]_9 $	Optimal	$ [[ 12, 9, 2; 1]]_9 $	No
$ Q_1 $	$ x^2 + x + 1 $	$ [15, 13, 2]_9 $	Optimal	$ [[ 15, 12, 2; 1]]_9 $	No
$ Q_1 $	$ x + 2 $	$ [15, 14, 2]_9 $	Optimal	$ [[ 15, 13, 2; 0]]_9 $	Yes
$ Q_2 $	$ x + 2 $	$ [15, 14, 2]_9 $	Optimal	$ [[ 15, 0, 15;13]]_9 $	Yes
$ Q_1 $	$ x + w^6 $	$ [16, 15, 2]_9 $	Optimal	$ [[ 16, 14, 2; 0]]_9 $	Yes
$ Q_2 $	$ x + w^6 $	$ [16, 15, 2]_9 $	Optimal	$ [[16, 0, 16; 14]]_9 $	Yes
$ Q_1 $	$ x^2 + w^6x + w^3 $	$ [16, 14, 2]_9 $	Optimal	$ [[ 16, 13, 2;1]]_9 $	No
$ Q_2 $	$ x^2 + w^6x + w^3 $	$ [16, 14, 2]_9 $	Optimal	$ [[16, 1, 14;13]]_9 $	No
$ Q_1 $	$ x^3 + 2x + w^2 $	$ [16, 13, 2]_9 $	Near Optimal	$ [[ 16, 12, 2; 2 ]]_9 $	No
$ Q_2 $	$ x^3 + 2x + w^2 $	$ [16, 13, 2]_9 $	Near Optimal	$ [[ 16, 2, 10;12]]_9 $	No
$ Q_1 $	$ x + 2 $	$ [18, 17, 2]_9 $	Optimal	$ [[ 18, 16, 2;0]]_9 $	Yes
$ Q_2 $	$ x + 2 $	$ [18, 17, 2]_9 $	Optimal	$ [[18, 0, 18; 16]]_9 $	Yes
$ Q_1 $	$ x^2 + w^3x + w^2 $	$ [20, 18, 2]_9 $	Optimal	$ [[ 20, 17, 2;1]]_9 $	No
$ Q_2 $	$ x^2 + w^3x + w^2 $	$ [20, 18, 2]_9 $	Optimal	$ [[20, 1, 15;17]]_9 $	No
$ Q_1 $	$ x + w^2 $	$ [20, 19, 2]_9 $	Optimal	$ [[ 20, 18, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + w^2 $	$ [20, 19, 2]_9 $	Optimal	$ [[20, 0, 20; 18]]_9 $	Yes
$ Q_1 $	$ x^3 + w^7x^2 + w^6x + w^6 $	$ [20, 17, 3]_9 $	Optimal	$ [[ 20, 16, 3; 2 ]]_9 $	No
$ Q_2 $	$ x^3 + w^7x^2 + w^6x + w^6 $	$ [20, 17, 3]_9 $	Optimal	$ [[20, 2, 16;16]]_9 $	No
$ Q_1 $	$ x^4 + w^6x^3 + wx^2 + w^2 $	$ [20, 16, 4]_9 $	Optimal	$ [[ 20, 15, 4; 3 ]]_9 $	No
$ Q_2 $	$ x^4 + w^6x^3 + wx^2 + w^2 $	$ [20, 16, 4]_9 $	Optimal	$ [[20, 3, 14; 15]]_9 $	No
$ Q_1 $	$ x^5 + w^5x^4 + wx^3 + w^3x^2 + w^6x + w^6 $	$ [20, 15, 4]_9 $	Near Optimal	$ [[ 20, 14, 4; 4 ]]_9 $	No
$ Q_2 $	$ x^5 + w^5x^4 + wx^3 + w^3x^2 + w^6x + w^6 $	$ [20, 15, 4]_9 $	Near Optimal	$ [[20, 4, 10; 14]]_9 $	No
$ Q_1 $	$ x + 2 $	$ [21, 20, 2]_9 $	Optimal	$ [[ 21, 19, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + 2 $	$ [21, 20, 2]_9 $	Optimal	$ [[21, 0, 21;19]]_9 $	Yes
$ Q_1 $	$ x^2 + x + 1 $	$ [21, 19, 2]_9 $	Optimal	$ [[ 21, 18, 2; 10 ]]_9 $	No
$ Q_2 $	$ x^2 + x + 1 $	$ [21, 19, 2]_9 $	Optimal	$ [[21, 1, 14;18]]_9 $	No
$ Q_1 $	$ x + w $	$ [24, 23, 2]_9 $	Optimal	$ [[ 24, 22, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + w $	$ [24, 23, 2]_9 $	Optimal	$ [[24, 0, 24; 22]]_9 $	Yes
$ Q_1 $	$ x^2 + x + 1 $	$ [24, 22, 2]_9 $	Optimal	$ [[ 24, 21, 2; 1 ]]_9 $	No
$ Q_2 $	$ x^2 + x + 1 $	$ [24, 22, 2]_9 $	Optimal	$ [[24, 1, 16; 21]]_9 $	No
$ Q_1 $	$ x^4 + wx^3 + 2x + w^5 $	$ [24, 20, 3]_9 $	Near Optimal	$ [[ 24, 19, 3; 3 ]]_9 $	No
$ Q_2 $	$ x^4 + wx^3 + 2x + w^5 $	$ [24, 20, 3]_9 $	Near Optimal	$ [[24, 3, 8; 19]]_9 $	No
$ Q_1 $	$ x + 2 $	$ [27, 26, 2]_9 $	Optimal	$ [[ 27, 25, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + 25 $	$ [27, 26, 2]_9 $	Optimal	$ [[27, 0, 27;25]]_9 $	Yes
$ Q_1 $	$ x + w^2 $	$ [28, 27, 2]_9 $	Optimal	$ [[ 28, 26, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + w^2 $	$ [28, 27, 2]_9 $	Optimal	$ [[28, 0, 28; 26]]_9 $	Yes
$ Q_1 $	$ x^2 + w^7x + w^2 $	$ [28, 26, 2]_9 $	Optimal	$ [[ 28, 25, 2; 1 ]]_9 $	No
$ Q_2 $	$ x^2 + w^7x + w^2 $	$ [28, 26, 2]_9 $	Optimal	$ [[28, 1, 21; 25]]_9 $	No
$ Q_1 $	$ x^3 + w^2x^2 + 2x + w^6 $	$ [28, 25, 2]_9 $	Near Optimal	$ [[28, 24, 2; 2 ]]_9 $	No
$ Q_2 $	$ x^3 + w^2x^2 + 2x + w^6 $	$ [28, 25, 2]_9 $	Near Optimal	$ [[28, 2, 14;24]]_9 $	No
$ Q_1 $	$ x + 1 $	$ [30, 29, 2]_9 $	Optimal	$ [[30, 28, 2; 0 ]]_9 $	Yes
$ Q_2 $	$ x + 1 $	$ [30, 29, 2]_9 $	Optimal	$ [[30, 0, 30; 28]]_9 $	Yes
$ Q_1 $	$ x^2 + 2*x + 1 $	$ [30, 28, 2]_9 $	Optimal	$ [[30, 27, 2; 1 ]]_9 $	No
$ Q_2 $	$ x^2 + 2*x + 1 $	$ [30, 28, 2]_9 $	Optimal	$ [[30, 1, 20; 27]]_9 $	No

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Table 3. one-dimensional linear complementary pairs of cyclic codes $ (\mathcal{C}, \mathcal{D}) $ over $ {\mathbb{F}}_{p^e} $

Alphabet	Parameters of $ \mathcal{C} $	Optimal/ Near Optimal	Parameters of $ \mathcal{D} $	Optimal/ Near Optimal
$ {\mathbb{F}}_{2} $	$ [12, 3, 6] $	Optimal	$ [12, 9, 2] $	Optimal
$ {\mathbb{F}}_{2} $	$ [14, 7, 4] $	Optimal	$ [14, 7, 4] $	Optimal
$ {\mathbb{F}}_{2} $	$ [20, 17, 2] $	Optimal	$ [20, 3, 10] $	Near Optimal
$ {\mathbb{F}}_{3} $	$ [6, 2, 4] $	Optimal	$ [6, 4, 2] $	Optimal
$ {\mathbb{F}}_{3} $	$ [6, 3, 2] $	Near Optimal	$ [6, 3, 3] $	Optimal
$ {\mathbb{F}}_{3} $	$ [8, 5, 3] $	Optimal	$ [8, 3, 5] $	Optimal
$ {\mathbb{F}}_{3} $	$ [12, 2, 8] $	Near Optimal	$ [12, 10, 2] $	Optimal
$ {\mathbb{F}}_{3} $	$ [16, 7, 5] $	Near Optimal	$ [16, 9, 4] $	Near Optimal
$ {\mathbb{F}}_{5} $	$ [4, 2, 3] $	Optimal	$ [4, 2, 3] $	Optimal
$ {\mathbb{F}}_{5} $	$ [8, 5, 2] $	Near Optimal	$ [8, 3, 4] $	Near Optimal
$ {\mathbb{F}}_{5} $	$ [8, 4, 4] $	Optimal	$ [8, 4, 3] $	Near Optimal
$ {\mathbb{F}}_{5} $	$ [8, 2, 6] $	Optimal	$ [8, 6, 2] $	Optimal
$ {\mathbb{F}}_{5} $	$ [8, 4, 4] $	Optimal	$ [8, 4, 4] $	Optimal
$ {\mathbb{F}}_{7} $	$ [6, 3, 3] $	Near Optimal	$ [6, 3, 3] $	Near Optimal
$ {\mathbb{F}}_{7} $	$ [6, 3, 4] $	Optimal	$ [6, 3, 3] $	Near Optimal
$ {\mathbb{F}}_{7} $	$ [6, 2, 4] $	Near Optimal	$ [6, 4, 2] $	Near Optimal
$ {\mathbb{F}}_{7} $	$ [6, 2, 5] $	Optimal	$ [6, 4, 2] $	Near Optimal
$ {\mathbb{F}}_{7} $	$ [8, 3, 6] $	Optimal	$ [8, 5, 4] $	Optimal
$ {\mathbb{F}}_{9} $	$ [4, 2, 3] $	Optimal	$ [4, 2, 3] $	Optimal
$ {\mathbb{F}}_{9} $	$ [4, 2, 2] $	Near Optimal	$ [4, 2, 3] $	Optimal
$ {\mathbb{F}}_{9} $	$ [6, 2, 4] $	Near Optimal	$ [6, 4, 2] $	Near Optimal
$ {\mathbb{F}}_{9} $	$ [8, 4, 5] $	Optimal	$ [8, 4, 4] $	Near Optimal
$ {\mathbb{F}}_{9} $	$ [8, 3, 5] $	Near Optimal	$ [8, 5, 4] $	Optimal
$ {\mathbb{F}}_{9} $	$ [8, 3, 6] $	Optimal	$ [8, 5, 3] $	Near Optimal
$ {\mathbb{F}}_{9} $	$ [8, 6, 3] $	Optimal	$ [8, 2, 6] $	Near Optimal
$ {\mathbb{F}}_{9} $	$ [8, 2, 7] $	Optimal	$ [8, 6, 2] $	Near Optimal

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