New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts

Adrian Korban; Serap Şahinkaya; Deniz Ustun

doi:10.3934/amc.2021034

Article Contents

2023, Volume 17, Issue 4: 994-1011. Doi: 10.3934/amc.2021034

This issue Previous Article Two classes of new optimal ternary cyclic codes Next Article

New type i binary [72, 36, 12] self-dual codes from composite matrices and R₁ lifts

1.
Department of Mathematical and Physical Sciences, University of Chester, Chester, England
2.
Faculty of Engineering, Department of Natural and Mathematical Sciences, Tarsus University, Mersin, Turkey
3.
Faculty of Engineering, Department of Computer Engineering, Tarsus University, Mersin, Turkey

^* Corresponding author: Adrian Korban
^* Corresponding author: Adrian Korban

Received: June 2020

Revised: June 2021

Early access: August 2021

Published: August 2023

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Abstract

In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form $ [I_n \ | \ \Omega(v)], $ where $ I_n $ is the identity matrix and $ \Omega(v) $ is a composite matrix and search for binary self-dual codes with parameters $ [36,18, 6 \ \text{or} \ 8]. $ We next lift these codes over the ring $ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $ to obtain codes whose binary images are self-dual codes with parameters $ [72,36,12]. $ Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find $ 30 $ new Type I binary self-dual codes with parameters $ [72,36,12]. $

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Mathematics Subject Classification: Primary: 94B05; Secondary: 16S34.

Citation:

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Table 1. Type I $ [36,18,6-8] $ Codes from Theorem 4.1

	Type	$ r_B $	$ r_C $	$ \|Aut(C_i)\| $
$ C_1 $	$ [36,18,6] $	$ (0,0,0,0,0,1,0,1,1) $	$ (1,0,1,1,1,0,1,0,1) $	$ 2^2 \cdot 3^2 $
$ C_2 $	$ [36,18,6] $	$ (0,0,0,0,1,1,0,1,1) $	$ (1,0,0,1,0,1,1,1,0) $	$ 2^2 \cdot 3^2 $
$ C_3 $	$ [36,18,8] $	$ (0,0,1,0,0,1,0,0,1) $	$ (1,0,0,1,1,0,1,1,1) $	$ 2^2 \cdot 3^2 $
$ C_4 $	$ [36,18,8] $	$ (0,1,0,0,0,1,0,1,1) $	$ (1,0,0,1,0,0,1,1,1) $	$ 2^2 \cdot 3^2 $

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Table 2. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_1 $

	Type	$ r_B $	$ r_C $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_1 $	$ W_{72,1} $	$ (u,0,u,u,u,1,u,u + 1,1) $	$ (1,0,1,1,u + 1,0,1,u,1) $	$ 0 $	$ 192 $	$ 36 $
$ \mathcal{C}_2 $	$ W_{72,1} $	$ (u,0,0,u,0,1,u,1,1) $	$ (1,0,u + 1,1,u + 1,0,1,u,u + 1) $	$ 0 $	$ 198 $	$ 36 $
$ \mathcal{C}_3 $	$ W_{72,1} $	$ (u,u,0,u,u,1,u,1,u + 1) $	$ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $	$ 0 $	$ 336 $	$ 36 $
$ \mathcal{C}_4 $	$ W_{72,1} $	$ (0,u,0,0,0,1,0,1,u + 1) $	$ (1,u,u + 1,1,u + 1,0,1,0,u + 1) $	$ 18 $	$ 234 $	$ 36 $
$ \mathcal{C}_5 $	$ W_{72,1} $	$ (u,u,0,u,0,1,u,u + 1,1) $	$ (1,u,u + 1,1,1,u,1,0,u + 1) $	$ 18 $	$ 345 $	$ 36 $
$ \mathcal{C}_6 $	$ W_{72,1} $	$ (0,u,0,0,0,1,0,u + 1,1) $	$ (1,u,1,1,u + 1,u,1,u,1) $	$ 18 $	$ 378 $	$ 36 $
$ \mathcal{C}_7 $	$ W_{72,1} $	$ (u,u,u,u,0,1,u,u + 1,u + 1) $	$ (1,u,1,1,u + 1,0,1,0,1) $	$ 18 $	$ 396 $	$ 36 $
$ \mathcal{C}_8 $	$ W_{72,1} $	$ (0,0,u,0,u,1,0,1,u + 1) $	$ (1,u,1,1,1,0,1,u,1) $	$ 18 $	$ 441 $	$ 36 $
$ \mathcal{C}_9 $	$ W_{72,1} $	$ (u,u,0,u,0,1,u,1,u + 1) $	$ (1,u,1,1,1,0,1,u,1) $	$ 18 $	$ 453 $	$ 36 $

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Table 3. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_2 $

	Type	$ r_B $	$ r_C $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_{10} $	$ W_{72,1} $	$ (0,u,0,0,1,1,0,1,u + 1) $	$ (1,u,u,1,u,u + 1,1,1,0) $	$ 0 $	$ 219 $	$ 36 $
$ \mathcal{C}_{11} $	$ W_{72,1} $	$ (u,0,u,u,1,1,u,u + 1,u + 1) $	$ (1,u,0,1,0,1,1,u + 1,u) $	$ 0 $	$ 345 $	$ 36 $
$ \mathcal{C}_{12} $	$ W_{72,1} $	$ (0,0,u,0,1,1,0,1,u + 1) $	$ (1,u,0,1,0,1,1,1,0) $	$ 0 $	$ 408 $	$ 36 $
$ \mathcal{C}_{13} $	$ W_{72,1} $	$ (u,0,0,u,1,u + 1,u,u + 1,u + 1) $	$ (1,0,u,1,0,u + 1,1,u + 1,0) $	$ 18 $	$ 261 $	$ 36 $
$ \mathcal{C}_{14} $	$ W_{72,1} $	$ (u,u,u,u,1,1,u,1,u + 1) $	$ (1,0,0,1,u,1,1,u + 1,0) $	$ 18 $	$ 270 $	$ 36 $
$ \mathcal{C}_{15} $	$ W_{72,1} $	$ (u,0,u,u,1,u + 1,u,1,u + 1) $	$ (1,0,u,1,u,1,1,u + 1,0) $	$ 18 $	$ 357 $	$ 36 $

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Table 4. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_3 $

	Type	$ r_B $	$ r_C $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_{16} $	$ W_{72,1} $	$ (u,u,1,u,0,1,u,0,1) $	$ (1,u,0,1,u + 1,u,1,1,u + 1) $	$ 0 $	$ 120 $	$ 36 $
$ \mathcal{C}_{17} $	$ W_{72,1} $	$ (u,u,1,u,0,1,u,u,1) $	$ (1,0,u,1,1,0,1,1,u + 1) $	$ 0 $	$ 282 $	$ 36 $
$ \mathcal{C}_{18} $	$ W_{72,1} $	$ (u,u,1,u,0,u + 1,u,0,1) $	$ (1,u,0,1,u + 1,u,1,1,u + 1) $	$ 0 $	$ 300 $	$ 36 $
$ \mathcal{C}_{19} $	$ W_{72,1} $	$ (u,u,1,u,0,u + 1,u,0,1) $	$ (1,u,u,1,1,0,1,1,1) $	$ 18 $	$ 336 $	$ 36 $
$ \mathcal{C}_{20} $	$ W_{72,1} $	$ (u,0,1,u,0,1,u,u,1) $	$ (1,0,0,1,1,0,1,u + 1,u + 1) $	$ 36 $	$ 435 $	$ 36 $

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Table 5. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_4 $

	Type	$ r_B $	$ r_C $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_{21} $	$ W_{72,1} $	$ (0,1,u,0,u,1,0,u + 1,u + 1) $	$ (1,u,u,1,u,u,1,u + 1,u + 1) $	$ 0 $	$ 366 $	$ 36 $
$ \mathcal{C}_{22} $	$ W_{72,1} $	$ (u,1,u,u,u,1,u,1,u + 1) $	$ (1,u,0,1,0,0,1,u + 1,1) $	$ 0 $	$ 372 $	$ 36 $
$ \mathcal{C}_{23} $	$ W_{72,1} $	$ (0,1,u,0,u,1,0,u + 1,u + 1) $	$ (1,0,0,1,0,0,1,u + 1,u + 1) $	$ 0 $	$ 384 $	$ 36 $
$ \mathcal{C}_{24} $	$ W_{72,1} $	$ (u,1,u,u,u,1,u,1,u + 1) $	$ (1,u,u,1,u,0,1,u + 1,1) $	$ 0 $	$ 390 $	$ 36 $
$ \mathcal{C}_{25} $	$ W_{72,1} $	$ (u,1,0,u,0,1,u,u + 1,1) $	$ (1,u,0,1,0,0,1,1,u + 1) $	$ 0 $	$ 399 $	$ 36 $
$ \mathcal{C}_{26} $	$ W_{72,1} $	$ (u,1,u,u,u,1,u,u + 1,1) $	$ (1,0,u,1,0,0,1,u + 1,u + 1) $	$ 18 $	$ 264 $	$ 36 $
$ \mathcal{C}_{27} $	$ W_{72,1} $	$ (u,1,0,u,u,u + 1,u,u + 1,1) $	$ (1,0,u,1,0,u,1,1,u + 1) $	$ 18 $	$ 285 $	$ 36 $
$ \mathcal{C}_{28} $	$ W_{72,1} $	$ (0,1,u,0,0,u + 1,0,1,1) $	$ (1,u,u,1,u,0,1,1,1) $	$ 18 $	$ 300 $	$ 36 $

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Table 6. Type I $ [36,18,6-8] $ Codes from Theorem 4.2

	Type	$ r_B $	$ r_C $	$ r_D $	$ \|Aut(C_i)\| $
$ C_5 $	$ [36,18,6] $	$ (0,0,0,0,0,1) $	$ (1,1,1,0,0,1) $	$ (1,1,1,0,1,0) $	$ 2^5 \cdot 3^4 \cdot 5 $
$ C_6 $	$ [36,18,6] $	$ (0,0,0,0,1,1) $	$ (0,0,0,0,1,1) $	$ (1,1,1,1,0,1) $	$ 2^5 \cdot 3^4 \cdot 5 $
$ C_7 $	$ [36,18,6] $	$ (0,0,0,0,1,1) $	$ (0,1,1,0,1,1) $	$ (0,1,1,1,0,0) $	$ 2^5 \cdot 3^2 $
$ C_8 $	$ [36,18,6] $	$ (0,0,1,0,0,1) $	$ (0,0,1,1,1,0) $	$ (1,1,1,0,0,1) $	$ 2^5 \cdot 3^2 $

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Table 7. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_7 $

	Type	$ r_B $	$ r_C $	$ r_D $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_{29} $	$ W_{72,1} $	$ (0,0,0,u,1,1) $	$ (u,1,u + 1,u,1,1) $	$ (u,u + 1,1,u + 1,0,0) $	$ 0 $	$ 471 $	$ 144 $

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Table 8. Type I $ [36,18,6-8] $ Codes from Theorem 4.3

	Type	$ r_B $	$ r_C $	$ r_D $	$ \|Aut(C_i)\| $
$ C_9 $	$ [36,18,6] $	$ (0,0,0,0,0,1) $	$ (0,1,1,0,1,1) $	$ (1,0,1,1,0,1) $	$ 2^5 \cdot 3^2 $
$ C_{10} $	$ [36,18,6] $	$ (0,0,0,0,0,1) $	$ (1,1,1,0,0,1) $	$ (1,1,1,0,1,0) $	$ 2^5 \cdot 3^4 \cdot 5 $
$ C_{11} $	$ [36,18,6] $	$ (0,0,0,0,1,1) $	$ (0,0,0,0,1,1) $	$ (1,1,1,1,0,1) $	$ 2^5 \cdot 3^4 \cdot 5 $
$ C_{12} $	$ [36,18,6] $	$ (0,0,0,0,1,1) $	$ (0,1,1,0,0,1) $	$ (1,1,0,1,0,1) $	$ 2^5 \cdot 3^2 $

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Table 9. New Type I $ [72,36,12] $ Codes from $ R_1 $-lift of $ C_9 $

	Type	$ r_B $	$ r_C $	$ r_D $	$ \gamma $	$ \beta $	$ \|Aut(\mathcal{C}_i)\| $
$ \mathcal{C}_{30} $	$ W_{72,1} $	$ (0,u,u,u,u,1) $	$ (u,1,1,u,u + 1,1) $	$ (u + 1,u,u + 1,1,u,u + 1) $	$ 0 $	$ 621 $	$ 432 $

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