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On the classification of $\mathbb{Z}_4$-codes
1. | Department of Computer Science, Shizuoka University, Hamamatsu 432-8011, Japan |
2. | Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan |
3. | Koki Consultant Inc. Kitakata 966-0902, Japan |
In this note, we study the classification of $\mathbb{Z}_4$-codes. For some special cases $(k_1,k_2)$, by hand, we give a classification of $\mathbb{Z}_4$-codes of length $n$ and type $4^{k_1}2^{k_2}$ satisfying a certain condition. Our exhaustive computer search completes the classification of $\mathbb{Z}_4$-codes of lengths up to $7$.
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
J. H. Conway and N. J. A. Sloane,
Self-dual codes over the integers modulo 4, J. Combin.Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[3] |
S. T. Dougherty, T. A. Gulliver, Y. H. Park and J. N. C. Wong,
Optimal linear codes over $\mathbb{Z}_m$, J. Korean Math. Soc., 44 (2007), 1139-1162.
doi: 10.4134/JKMS.2007.44.5.1139. |
[4] |
T. A. Gulliver and J. N. C. Wong,
Classification of optimal linear $\mathbb{Z}_4$ rate $1/2$ codes of length $≤ 8$, Ars Combin., 85 (2007), 287-306.
|
[5] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. 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. |
[6] |
M. Harada and A. Munemasa,
On the classification of self-dual $\mathbb{Z}_k$-codes, Lecture Notes in Comput. Sci., 5921 (2009), 78-90.
|
[7] |
J. N. C. Wong,
Classification of Small Optimal Linear Codes Over $Z_4$, Master's thesis, University of Victoria, 2002. |
show all references
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
J. H. Conway and N. J. A. Sloane,
Self-dual codes over the integers modulo 4, J. Combin.Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[3] |
S. T. Dougherty, T. A. Gulliver, Y. H. Park and J. N. C. Wong,
Optimal linear codes over $\mathbb{Z}_m$, J. Korean Math. Soc., 44 (2007), 1139-1162.
doi: 10.4134/JKMS.2007.44.5.1139. |
[4] |
T. A. Gulliver and J. N. C. Wong,
Classification of optimal linear $\mathbb{Z}_4$ rate $1/2$ codes of length $≤ 8$, Ars Combin., 85 (2007), 287-306.
|
[5] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. 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. |
[6] |
M. Harada and A. Munemasa,
On the classification of self-dual $\mathbb{Z}_k$-codes, Lecture Notes in Comput. Sci., 5921 (2009), 78-90.
|
[7] |
J. N. C. Wong,
Classification of Small Optimal Linear Codes Over $Z_4$, Master's thesis, University of Victoria, 2002. |
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1 | 1 | 7 |
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| 0 | 3 | 1 | 3 | 0 | 1 | |
1 | 1 | 7 |
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| | 1 | | 1 | 3 | 4 | |
| | | 3 | 2 | 1 | 23 | |
| | 4 | | 2 | 2 | 6 | |
| 0 | 3 | 3 | 3 | 0 | 9 | |
1 | 1 | 17 | | 3 | 1 | 4 | |
| 0 | 4 | 1 | | 4 | 0 | 1 |
1 | 2 | 16 | |||||
2 | 0 | 18 |
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| | 1 | | 1 | 3 | 4 | |
| | | 3 | 2 | 1 | 23 | |
| | 4 | | 2 | 2 | 6 | |
| 0 | 3 | 3 | 3 | 0 | 9 | |
1 | 1 | 17 | | 3 | 1 | 4 | |
| 0 | 4 | 1 | | 4 | 0 | 1 |
1 | 2 | 16 | |||||
2 | 0 | 18 |
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| | | 4 | 2 | 2 | 67 | |
| | 5 | 3 | 0 | 63 | ||
| 0 | 3 | 6 | 2 | 3 | 10 | |
1 | 1 | 33 | 3 | 1 | 55 | ||
| 0 | 4 | 4 | 3 | 2 | 10 | |
1 | 2 | 54 | 4 | 0 | 14 | ||
2 | 0 | 49 | 4 | 1 | 5 | ||
| 0 | 5 | 1 | 5 | 0 | 1 | |
1 | 3 | 29 | |||||
2 | 1 | 121 |
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| | 1 | 1 | 4 | 5 | ||
| | | 4 | 2 | 2 | 67 | |
| | 5 | 3 | 0 | 63 | ||
| 0 | 3 | 6 | 2 | 3 | 10 | |
1 | 1 | 33 | 3 | 1 | 55 | ||
| 0 | 4 | 4 | 3 | 2 | 10 | |
1 | 2 | 54 | 4 | 0 | 14 | ||
2 | 0 | 49 | 4 | 1 | 5 | ||
| 0 | 5 | 1 | 5 | 0 | 1 | |
1 | 3 | 29 | |||||
2 | 1 | 121 |
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| | 1 | | 1 | 5 | 6 | |
| | | 6 | 2 | 3 | 157 | |
| | 6 | 3 | 1 | 587 | ||
| 0 | 3 | 12 | | 2 | 4 | 16 |
1 | 1 | 58 | 3 | 2 | 212 | ||
| 0 | 4 | 11 | 4 | 0 | 179 | |
1 | 2 | 149 | | 3 | 3 | 22 | |
2 | 0 | 121 | 4 | 1 | 112 | ||
| 0 | 5 | 5 | | 4 | 2 | 16 |
1 | 3 | 134 | | 0 | 20 | ||
2 | 1 | 499 | | 5 | 1 | 6 | |
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1 | 4 | 47 | |||||
2 | 2 | 500 | |||||
3 | 0 | 381 |
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| | | 6 | 2 | 3 | 157 | |
| | 6 | 3 | 1 | 587 | ||
| 0 | 3 | 12 | | 2 | 4 | 16 |
1 | 1 | 58 | 3 | 2 | 212 | ||
| 0 | 4 | 11 | 4 | 0 | 179 | |
1 | 2 | 149 | | 3 | 3 | 22 | |
2 | 0 | 121 | 4 | 1 | 112 | ||
| 0 | 5 | 5 | | 4 | 2 | 16 |
1 | 3 | 134 | | 0 | 20 | ||
2 | 1 | 499 | | 5 | 1 | 6 | |
| 0 | 6 | 1 | | 6 | 0 | 1 |
1 | 4 | 47 | |||||
2 | 2 | 500 | |||||
3 | 0 | 381 |
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| | 1 | 1 | 6 | 7 | ||
| | | 7 | 2 | 4 | 319 | |
| | 7 | 3 | 2 | 3247 | ||
| 0 | 3 | 21 | 4 | 0 | 2215 | |
1 | 1 | 93 | 2 | 5 | 23 | ||
| 0 | 4 | 27 | 3 | 3 | 648 | |
1 | 2 | 359 | 4 | 1 | 2257 | ||
2 | 0 | 256 | 3 | 4 | 43 | ||
| 0 | 5 | 17 | 4 | 2 | 565 | |
1 | 3 | 503 | 5 | 0 | 429 | ||
2 | 1 | 1728 | 4 | 3 | 43 | ||
| 0 | 6 | 6 | 5 | 1 | 204 | |
1 | 4 | 283 | 5 | 2 | 23 | ||
2 | 2 | 2896 | 6 | 0 | 27 | ||
3 | 0 | 1955 | 6 | 1 | 7 | ||
| 0 | 7 | 1 | 7 | 0 | 1 | |
1 | 5 | 70 | |||||
2 | 3 | 1582 | |||||
3 | 1 | 5184 |
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| | 1 | 1 | 6 | 7 | ||
| | | 7 | 2 | 4 | 319 | |
| | 7 | 3 | 2 | 3247 | ||
| 0 | 3 | 21 | 4 | 0 | 2215 | |
1 | 1 | 93 | 2 | 5 | 23 | ||
| 0 | 4 | 27 | 3 | 3 | 648 | |
1 | 2 | 359 | 4 | 1 | 2257 | ||
2 | 0 | 256 | 3 | 4 | 43 | ||
| 0 | 5 | 17 | 4 | 2 | 565 | |
1 | 3 | 503 | 5 | 0 | 429 | ||
2 | 1 | 1728 | 4 | 3 | 43 | ||
| 0 | 6 | 6 | 5 | 1 | 204 | |
1 | 4 | 283 | 5 | 2 | 23 | ||
2 | 2 | 2896 | 6 | 0 | 27 | ||
3 | 0 | 1955 | 6 | 1 | 7 | ||
| 0 | 7 | 1 | 7 | 0 | 1 | |
1 | 5 | 70 | |||||
2 | 3 | 1582 | |||||
3 | 1 | 5184 |
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