# American Institute of Mathematical Sciences

November 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 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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##### 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.
Length 1
 $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $2$ $0$ $1$ $1$ $2^2$ $1$ $0$ $1$
 $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(1,k_1,k_2)$ $2$ $0$ $1$ $1$ $2^2$ $1$ $0$ $1$
Length 2
 $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $2$ $0$ $1$ $1$ $2^3$ 1 1 2 $2^2$ $0$ $2$ $1$ $2^4$ 2 0 1 $1$ $0$ $2$
 $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(2,k_1,k_2)$ $2$ $0$ $1$ $1$ $2^3$ 1 1 2 $2^2$ $0$ $2$ $1$ $2^4$ 2 0 1 $1$ $0$ $2$
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 $2^4$ 1 2 3 $2^2$ $0$ $2$ 2 2 0 5 $1$ $0$ 3 $2^5$ 2 1 3 $2^3$ 0 3 1 $2^6$ 3 0 1 1 1 7
 $|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(3,k_1,k_2)$ $2$ $0$ $1$ 1 $2^4$ 1 2 3 $2^2$ $0$ $2$ 2 2 0 5 $1$ $0$ 3 $2^5$ 2 1 3 $2^3$ 0 3 1 $2^6$ 3 0 1 1 1 7
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 $2^5$ 1 3 4 $2^2$ $0$ $2$ 3 2 1 23 $1$ $0$ 4 $2^6$ 2 2 6 $2^3$ 0 3 3 3 0 9 1 1 17 $2^7$ 3 1 4 $2^4$ 0 4 1 $2^8$ 4 0 1 1 2 16 2 0 18
 $|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(4,k_1,k_2)$ $2$ $0$ $1$ 1 $2^5$ 1 3 4 $2^2$ $0$ $2$ 3 2 1 23 $1$ $0$ 4 $2^6$ 2 2 6 $2^3$ 0 3 3 3 0 9 1 1 17 $2^7$ 3 1 4 $2^4$ 0 4 1 $2^8$ 4 0 1 1 2 16 2 0 18
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 $2^6$ 1 4 5 $2^2$ $0$ $2$ 4 2 2 67 $1$ $0$ 5 3 0 63 $2^3$ 0 3 6 $2^7$ 2 3 10 1 1 33 3 1 55 $2^4$ 0 4 4 $2^8$ 3 2 10 1 2 54 4 0 14 2 0 49 $2^9$ 4 1 5 $2^5$ 0 5 1 $2^{10}$ 5 0 1 1 3 29 2 1 121
 $|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(5,k_1,k_2)$ $2$ $0$ $1$ 1 $2^6$ 1 4 5 $2^2$ $0$ $2$ 4 2 2 67 $1$ $0$ 5 3 0 63 $2^3$ 0 3 6 $2^7$ 2 3 10 1 1 33 3 1 55 $2^4$ 0 4 4 $2^8$ 3 2 10 1 2 54 4 0 14 2 0 49 $2^9$ 4 1 5 $2^5$ 0 5 1 $2^{10}$ 5 0 1 1 3 29 2 1 121
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 $2^7$ 1 5 6 $2^2$ $0$ $2$ 6 2 3 157 $1$ $0$ 6 3 1 587 $2^3$ 0 3 12 $2^8$ 2 4 16 1 1 58 3 2 212 $2^4$ 0 4 11 4 0 179 1 2 149 $2^9$ 3 3 22 2 0 121 4 1 112 $2^5$ 0 5 5 $2^{10}$ 4 2 16 1 3 134 $5$ 0 20 2 1 499 $2^{11}$ 5 1 6 $2^6$ 0 6 1 $2^{12}$ 6 0 1 1 4 47 2 2 500 3 0 381
 $|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(6,k_1,k_2)$ $2$ $0$ $1$ 1 $2^7$ 1 5 6 $2^2$ $0$ $2$ 6 2 3 157 $1$ $0$ 6 3 1 587 $2^3$ 0 3 12 $2^8$ 2 4 16 1 1 58 3 2 212 $2^4$ 0 4 11 4 0 179 1 2 149 $2^9$ 3 3 22 2 0 121 4 1 112 $2^5$ 0 5 5 $2^{10}$ 4 2 16 1 3 134 $5$ 0 20 2 1 499 $2^{11}$ 5 1 6 $2^6$ 0 6 1 $2^{12}$ 6 0 1 1 4 47 2 2 500 3 0 381
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 $2^8$ 1 6 7 $2^2$ $0$ $2$ 7 2 4 319 $1$ $0$ 7 3 2 3247 $2^3$ 0 3 21 4 0 2215 1 1 93 $2^9$ 2 5 23 $2^4$ 0 4 27 3 3 648 1 2 359 4 1 2257 2 0 256 $2^{10}$ 3 4 43 $2^5$ 0 5 17 4 2 565 1 3 503 5 0 429 2 1 1728 $2^{11}$ 4 3 43 $2^6$ 0 6 6 5 1 204 1 4 283 $2^{12}$ 5 2 23 2 2 2896 6 0 27 3 0 1955 $2^{13}$ 6 1 7 $2^7$ 0 7 1 $2^{14}$ 7 0 1 1 5 70 2 3 1582 3 1 5184
 $|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$ $|C|$ $k_1$ $k_2$ $N'(7,k_1,k_2)$ $2$ $0$ $1$ 1 $2^8$ 1 6 7 $2^2$ $0$ $2$ 7 2 4 319 $1$ $0$ 7 3 2 3247 $2^3$ 0 3 21 4 0 2215 1 1 93 $2^9$ 2 5 23 $2^4$ 0 4 27 3 3 648 1 2 359 4 1 2257 2 0 256 $2^{10}$ 3 4 43 $2^5$ 0 5 17 4 2 565 1 3 503 5 0 429 2 1 1728 $2^{11}$ 4 3 43 $2^6$ 0 6 6 5 1 204 1 4 283 $2^{12}$ 5 2 23 2 2 2896 6 0 27 3 0 1955 $2^{13}$ 6 1 7 $2^7$ 0 7 1 $2^{14}$ 7 0 1 1 5 70 2 3 1582 3 1 5184
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