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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 |
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.
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. |
[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. |
[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. |
[4] |
A. Bonisoli,
Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
C. Carlet, One-weight $\mathbb{Z}_{4}$-linear codes, Springer Berlin, (2000), 57–72. |
[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. |
[12] |
P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[19] |
Z. X. Wan, Quaternary Codes, Singapore, World Scientific, 1997.
doi: 10.1142/3603. |
[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. |
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. |
[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. |
[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. |
[4] |
A. Bonisoli,
Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
C. Carlet, One-weight $\mathbb{Z}_{4}$-linear codes, Springer Berlin, (2000), 57–72. |
[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. |
[12] |
P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[19] |
Z. X. Wan, Quaternary Codes, Singapore, World Scientific, 1997.
doi: 10.1142/3603. |
[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. |
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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 |
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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