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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

Jie Geng 1, , Huazhang Wu 1,, and  Patrick Solé 2,

1. 

School of Mathematical Sciences, Anhui University, Hefei 230601, China
2. 

I2M, (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, France

* Corresponding author: Huazhang Wu

Received  March 2021 Revised  July 2021 Early access October 2021

Fund Project: 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

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: Additive codes, Gray map, $ N $-weight codes, two-weight projective codes.
Mathematics Subject Classification: Primary: 94B05, 94B15, 94B60.
Citation: Jie Geng, Huazhang Wu, Patrick Solé. On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions. Advances in Mathematics of Communications, doi: 10.3934/amc.2021046
References:
[1]

T. Abualrub, I. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

[2]

I. Aydogdu, T. Abualrub and I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.  Google Scholar

[3]

I. Aydogdu and I. Siap, The structure of $\mathbb{Z}_{2}\mathbb{Z}_{2^{s}}$-additive codes: Bounds on the minimum distance, Appl. Math. Inf. Sci., 7 (2013), 2271-2278.  doi: 10.12785/amis/070617.  Google Scholar

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.   Google Scholar

[5]

J. Borges, C. Fernández-Cárdoba, J. Pujól, J. Rifà and M. Villanueva, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Generator matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.  Google Scholar

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J. Borges, C. Fernández-Cárdoba and R. Ten-Valls, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Info. Theory, 62 (2016), 6348-6354.  doi: 10.1109/TIT.2016.2611528.  Google Scholar

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I. Bouyuliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78.  doi: 10.1007/s10623-006-0019-1.  Google Scholar

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A. E. Brouwer, Some new two-weight codes and strongly regular graphs, Discrete Appl. Math., 10 (1985), 111-114.  doi: 10.1016/0166-218X(85)90062-9.  Google Scholar

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R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[10]

C. Carlet, One-weight $\mathbb{Z}_{4}$-linear codes, Springer Berlin, (2000), 57–72.  Google Scholar

[11]

F. D. Clerck and M. Delanote, Two-weight codes, partial geometries and Steiner systems, Des. Codes Cryptogr., 21 (2000), 87-98.  doi: 10.1023/A:1008383510488.  Google Scholar

[12]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973.  Google Scholar

[13]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. Sloane and P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[14]

H. Rifà, J. Rifà and L. Ronquilloy, Perfect $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes in steganography, Comput. Res. Reposit, 26 (2010), 696-701.   Google Scholar

[15]

J. Rifà and L. Ronquillo, Product perfect $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes in steganography, International Symposium on Information Theory & Its Applications, (2010), 17–20. Google Scholar

[16]

M. Sari, V. Siap and I. Siap, One-homogeneous weight codes over finite chain rings, Bull. Korean Math. Soc., 52 (2015), 2011-2023.  doi: 10.4134/BKMS.2015.52.6.2011.  Google Scholar

[17]

M. J. Shi, C. C. Wang, R. S. Wu, Y. Hu and Y. Q. Chang, One-weight and two-weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u, v]$-additive codes, Cryptogr. Commun., 12 (2020), 443-454.  doi: 10.1007/s12095-019-00391-5.  Google Scholar

[18]

M. J. Shi, L. L. Xu and G. Yang, A note on one weight and two weight projective $\mathbb{Z}_{4}$-codes, IEEE Trans. Inf. Theory, 63 (2017), 177-182.  doi: 10.1109/TIT.2016.2628408.  Google Scholar

[19]

Z. X. Wan, Quaternary Codes, Singapore, World Scientific, 1997. doi: 10.1142/3603.  Google Scholar

[20]

J. A. Wood, The structure of linear codes of constant weight, Trans. Amer. Math. Soc., 354 (2002), 1007-1026.  doi: 10.1090/S0002-9947-01-02905-1.  Google Scholar

show all references

References:
[1]

T. Abualrub, I. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

[2]

I. Aydogdu, T. Abualrub and I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.  Google Scholar

[3]

I. Aydogdu and I. Siap, The structure of $\mathbb{Z}_{2}\mathbb{Z}_{2^{s}}$-additive codes: Bounds on the minimum distance, Appl. Math. Inf. Sci., 7 (2013), 2271-2278.  doi: 10.12785/amis/070617.  Google Scholar

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.   Google Scholar

[5]

J. Borges, C. Fernández-Cárdoba, J. Pujól, J. Rifà and M. Villanueva, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Generator matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.  Google Scholar

[6]

J. Borges, C. Fernández-Cárdoba and R. Ten-Valls, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Info. Theory, 62 (2016), 6348-6354.  doi: 10.1109/TIT.2016.2611528.  Google Scholar

[7]

I. Bouyuliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78.  doi: 10.1007/s10623-006-0019-1.  Google Scholar

[8]

A. E. Brouwer, Some new two-weight codes and strongly regular graphs, Discrete Appl. Math., 10 (1985), 111-114.  doi: 10.1016/0166-218X(85)90062-9.  Google Scholar

[9]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[10]

C. Carlet, One-weight $\mathbb{Z}_{4}$-linear codes, Springer Berlin, (2000), 57–72.  Google Scholar

[11]

F. D. Clerck and M. Delanote, Two-weight codes, partial geometries and Steiner systems, Des. Codes Cryptogr., 21 (2000), 87-98.  doi: 10.1023/A:1008383510488.  Google Scholar

[12]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973.  Google Scholar

[13]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. Sloane and P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[14]

H. Rifà, J. Rifà and L. Ronquilloy, Perfect $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes in steganography, Comput. Res. Reposit, 26 (2010), 696-701.   Google Scholar

[15]

J. Rifà and L. Ronquillo, Product perfect $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes in steganography, International Symposium on Information Theory & Its Applications, (2010), 17–20. Google Scholar

[16]

M. Sari, V. Siap and I. Siap, One-homogeneous weight codes over finite chain rings, Bull. Korean Math. Soc., 52 (2015), 2011-2023.  doi: 10.4134/BKMS.2015.52.6.2011.  Google Scholar

[17]

M. J. Shi, C. C. Wang, R. S. Wu, Y. Hu and Y. Q. Chang, One-weight and two-weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u, v]$-additive codes, Cryptogr. Commun., 12 (2020), 443-454.  doi: 10.1007/s12095-019-00391-5.  Google Scholar

[18]

M. J. Shi, L. L. Xu and G. Yang, A note on one weight and two weight projective $\mathbb{Z}_{4}$-codes, IEEE Trans. Inf. Theory, 63 (2017), 177-182.  doi: 10.1109/TIT.2016.2628408.  Google Scholar

[19]

Z. X. Wan, Quaternary Codes, Singapore, World Scientific, 1997. doi: 10.1142/3603.  Google Scholar

[20]

J. A. Wood, The structure of linear codes of constant weight, Trans. Amer. Math. Soc., 354 (2002), 1007-1026.  doi: 10.1090/S0002-9947-01-02905-1.  Google Scholar

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