On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

Jie Geng; Huazhang Wu; Patrick Solé

doi:10.3934/amc.2021046

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2021, Volume 0. Doi: 10.3934/amc.2021046

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On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

1.
School of Mathematical Sciences, Anhui University, Hefei 230601, China
2.
I2M, (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, France

^* Corresponding author: Huazhang Wu
^* Corresponding author: Huazhang Wu

Received: February 28, 2021

Revised: June 30, 2021

Early access: October 2021

The research is supported by the Open Fund Research of Fund of Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University

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Abstract

This paper mainly study $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes. A Gray map from $ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $ to $ \mathbb{Z}_{4}^{\alpha+2\beta} $ is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive code and its dual is proved. Some properties of one-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes and two-weight projective $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are discussed. As main results, some construction methods for one-weight and two-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are studied, meanwhile several examples are presented to illustrate the methods.

Keywords:

Mathematics Subject Classification: Primary: 94B05, 94B15, 94B60.

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Table 1. One-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes

Cases	Weight	Remark
$ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0 $	$ 8k_{1} $	$ a+4a_{7}=2k_{1} $
$ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L} ((2+u)\mathbf{c})=0 $	$ 4k_{3} $	$ a+4a_{7}=2k_{3} $
$ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0 $	$ 4k_{2} $	$ a+4a_{7}=2k_{2} $
$ w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0 $	$ 4k_{1} $	$ a+4a_{7}=2k_{1} $
$ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})=0 $	/	/
$ w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L} ((2 +u)\mathbf{c})=0 $	/	/
$ w_{L}(\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=0 $	/	/
$ w_{L}(\mathbf{c})\neq0,w_{L} ((2 +u)\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})=0 $	$ a+4a_{7} $	/

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Table 2. Two-weight $\mathbb{Z}_{2}\mathbb{Z}_{4}[u]$-additive codes

Cases	$m_{1}$	$m_{2}$	Remark
$w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0$	$4(k_{1}+k_{3})$	$8k_{1}$	$a=2k_{3}-4a_{7},k_{1}\neq k_{3}$
$w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0$	$4(k_{1}+k_{2})$	$8k_{1}$	$a=2k_{2}-4a_{7},k_{1}\neq k_{2}$
$w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0$	$4(k_{1}+k_{2})$	$8k_{2}$	$a=2k_{1}-4a_{7},k_{1}\neq k_{2}$
$w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0$	$a+6k_{1}+4a_{7}$	$8k_{1}$	$a+4a_{7}\neq2k_{1}$
$w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0$	$4(k_{1}+k_{2})$	$8k_{2}$	$a=4k_{2}-2k_{1}-4a_{7},k_{1}\neq k_{2}$
$w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0$	$4(k_{1}+k_{2})$	$8k_{1}$	$a=4k_{1}-2k_{2}-4a_{7},k_{1}\neq k_{2}$
$w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})\neq0$	$4(k_{1}+k_{3})$	$8k_{1}$	$a=4k_{1}-2k_{3}-4a_{7},k_{1}\neq k_{3}$
$w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$	/	/	/
$w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$	/	/	/
$w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$	$a+2k_{3}+4a_{7}$	$4k_{3}$	$a+4a_{7}\neq2k_{3}$
$w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$	/	/	/
$w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$	/	/	/
$w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$	$a+2k_{2}+4a_{7}$	$4k_{2}$	$a+4a_{7}\neq2k_{2}$
$w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$	/	/	/
$w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$	/	/	/
$w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$	$a+2k_{1}+4a_{7}$	$4k_{1}$	$a+4a_{7}\neq2k_{1}$
$w_{L}(\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})=0$	/	/	/
$w_{L}(\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})=0$	/	/	/
$w_{L}(\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})=0$	/	/	/

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Table 3. Code parameters comparison

Examples	Length of $\Phi(\mathcal{C})$	Size of $\Phi(\mathcal{C})$	Lee weight of $\Phi(\mathcal{C})$	Lee weight in Database in http://www.Z4codes.info/	Remark
Ex. 5.3 (i)	8	4	8	8/10	As good as in Database
Ex. 5.3 (ii)	10	2	12	/	New value
Ex. 5.5 (i)	32	4	32	32/42	As good as in Database
Ex. 5.5 (ii)	30	4	32	30/40	Better than Database
Ex. 5.5 (iii)	62	4	64	82
Ex. 5.7	36	8	32	/	New value
Ex. 6.2 (i)	9	4	6 and 12	9/12	Optimal as per Database
Ex. 6.2 (ii)	16	4	8 and 16	16/21	Optimal as per Database
Ex. 6.4 (i)	18	4	16 and 20	18/24	Improves on Database
Ex. 6.4 (ii)	11	4	12 and 13	11/14	Improves on Database
Ex. 6.6	46	8	36 and 48	/	New value

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