On the classification of $\mathbb{Z}_4$-codes

Makoto Araya; Masaaki Harada; Hiroki Ito; Ken Saito

doi:10.3934/amc.2017054

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

On the classification of $\mathbb{Z}_4$-codes

Makoto Araya ^1, , Masaaki Harada ^2, , Hiroki Ito ^3, and Ken Saito ^2,

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 $7$.

Keywords: .

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B25.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. N. C. Wong, Classification of Small Optimal Linear Codes Over $Z_4$, Master's thesis, University of Victoria, 2002. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. N. C. Wong, Classification of Small Optimal Linear Codes Over $Z_4$, Master's thesis, University of Victoria, 2002. Google Scholar

Table 1. Length 1

$\|C\|$	$k_1$	$k_2$	$N'(1,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(1,k_1,k_2)$
$2$	$0$	$1$	$1$	$2^2$	$1$	$0$	$1$

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$\|C\|$	$k_1$	$k_2$	$N'(1,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(1,k_1,k_2)$
$2$	$0$	$1$	$1$	$2^2$	$1$	$0$	$1$

Table 2. Length 2

$\|C\|$	$k_1$	$k_2$	$N'(2,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(2,k_1,k_2)$
$2$	$0$	$1$	$1$	$2^3$	1	1	2
$2^2$	$0$	$2$	$1$	$2^4$	2	0	1
	$1$	$0$	$2$

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$\|C\|$	$k_1$	$k_2$	$N'(2,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(2,k_1,k_2)$
$2$	$0$	$1$	$1$	$2^3$	1	1	2
$2^2$	$0$	$2$	$1$	$2^4$	2	0	1
	$1$	$0$	$2$

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	$2^4$	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

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$\|C\|$	$k_1$	$k_2$	$N'(3,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(3,k_1,k_2)$
$2$	$0$	$1$	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

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	$2^5$	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

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$\|C\|$	$k_1$	$k_2$	$N'(4,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(4,k_1,k_2)$
$2$	$0$	$1$	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

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	$2^6$	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

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$\|C\|$	$k_1$	$k_2$	$N'(5,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(5,k_1,k_2)$
$2$	$0$	$1$	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

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	$2^7$	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

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$\|C\|$	$k_1$	$k_2$	$N'(6,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(6,k_1,k_2)$
$2$	$0$	$1$	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

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	$2^8$	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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$\|C\|$	$k_1$	$k_2$	$N'(7,k_1,k_2)$	$\|C\|$	$k_1$	$k_2$	$N'(7,k_1,k_2)$
$2$	$0$	$1$	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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