2017, 11(4): 747-756. doi: 10.3934/amc.2017054

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

Received  April 2016 Published  November 2017

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 $7$.

Citation: Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
References:
[1]

W. Bosma, J. Cannon, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

Table 1.  Length 1
$|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$
$2$ $0$ $1$ $1$ $2^2$ $1$ $0$ $1$
$|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$
$2$ $0$ $1$ $1$ $2^2$ $1$ $0$ $1$
Table 2.  Length 2
$|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$
$2$ $0$ $1$ $1$ $2^3$112
$2^2$ $0$ $2$ $1$ $2^4$201
$1$ $0$ $2$
$|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$
$2$ $0$ $1$ $1$ $2^3$112
$2^2$ $0$ $2$ $1$ $2^4$201
$1$ $0$ $2$
Table 3.  Length 3
$|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$
$2$ $0$ $1$1$2^4$123
$2^2$ $0$ $2$2205
$1$ $0$3$2^5$213
$2^3$031$2^6$301
117
$|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$
$2$ $0$ $1$1$2^4$123
$2^2$ $0$ $2$2205
$1$ $0$3$2^5$213
$2^3$031$2^6$301
117
Table 4.  Length 4
$|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$
$2$ $0$ $1$1 $2^5$134
$2^2$ $0$ $2$32123
$1$ $0$4 $2^6$226
$2^3$033309
1117 $2^7$314
$2^4$041 $2^8$401
1216
2018
$|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$
$2$ $0$ $1$1 $2^5$134
$2^2$ $0$ $2$32123
$1$ $0$4 $2^6$226
$2^3$033309
1117 $2^7$314
$2^4$041 $2^8$401
1216
2018
Table 5.  Length 5
$|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$
$2$ $0$ $1$1$2^6$145
$2^2$ $0$ $2$42267
$1$ $0$53063
$2^3$036$2^7$2310
11333155
$2^4$044$2^8$3210
12544014
2049$2^9$415
$2^5$051$2^{10}$501
1329
21121
$|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$
$2$ $0$ $1$1$2^6$145
$2^2$ $0$ $2$42267
$1$ $0$53063
$2^3$036$2^7$2310
11333155
$2^4$044$2^8$3210
12544014
2049$2^9$415
$2^5$051$2^{10}$501
1329
21121
Table 6.  Length 6
$|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$
$2$ $0$ $1$1 $2^7$156
$2^2$ $0$ $2$623157
$1$ $0$631587
$2^3$0312 $2^8$2416
115832212
$2^4$041140179
12149 $2^9$3322
2012141112
$2^5$055 $2^{10}$4216
13134 $5$020
21499 $2^{11}$516
$2^6$061 $2^{12}$601
1447
22500
30381
$|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$
$2$ $0$ $1$1 $2^7$156
$2^2$ $0$ $2$623157
$1$ $0$631587
$2^3$0312 $2^8$2416
115832212
$2^4$041140179
12149 $2^9$3322
2012141112
$2^5$055 $2^{10}$4216
13134 $5$020
21499 $2^{11}$516
$2^6$061 $2^{12}$601
1447
22500
30381
Table 7.  Length 7
$|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$
$2$ $0$ $1$1$2^8$167
$2^2$ $0$ $2$724319
$1$ $0$7323247
$2^3$0321402215
1193$2^9$2523
$2^4$042733648
12359412257
20256$2^{10}$3443
$2^5$051742565
1350350429
211728$2^{11}$4343
$2^6$06651204
14283$2^{12}$5223
2228966027
301955$2^{13}$617
$2^7$071$2^{14}$701
1570
231582
315184
$|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$
$2$ $0$ $1$1$2^8$167
$2^2$ $0$ $2$724319
$1$ $0$7323247
$2^3$0321402215
1193$2^9$2523
$2^4$042733648
12359412257
20256$2^{10}$3443
$2^5$051742565
1350350429
211728$2^{11}$4343
$2^6$06651204
14283$2^{12}$5223
2228966027
301955$2^{13}$617
$2^7$071$2^{14}$701
1570
231582
315184
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