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

• 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 $7$ .

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

 Citation:

• Table 1.  Length 1

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

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 $1$ $0$ $2$

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

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

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

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

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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