February  2017, 11(1): 99-114. doi: 10.3934/amc.2017005

Cyclic codes over local Frobenius rings of order 16

1. 

Department of Mathematics, University of Scranton, Scranton, PA 18518, USA

2. 

Department of Mathematics, Fatih University, Istanbul, 34500, Turkey

3. 

Department of Mathematics, University of Scranton, Scranton, PA 18518, USA

* Corresponding author

Received  June 2015 Published  February 2017

Fund Project: The third author would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper.

We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techniques including new optimal quasi-cyclic codes.

Citation: Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
References:
[1]

N. Aydin and T. Asamov, A database of $\mathbb Z_4$-codes, available at http://www.asamov.com/Z4Codes. Google Scholar

[2]

A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb{F}_2 + u\mathbb{F}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1109/18.761278.  Google Scholar

[3]

A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptpgr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.  Google Scholar

[4]

E. Z. Chen, A database of quasi-twisted codes, available at http://moodle.tec.hkr.se/~chen/research/codes. Google Scholar

[5]

H. Q. Dinh and S. Lopez-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

[6]

S. T. DoughertyS. Karadeniz and B. Yildiz, Cyclic codes over Rk, Des. Codes Cryptogr., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.  Google Scholar

[7]

S. T. DoughertyE. Salturk and S. Szabo, Codes over local rings of order 16 and binary codes, Adv. math. Comm., 10 (), 379-391.  doi: 10.3934/amc.2016012.  Google Scholar

[8]

S. T. Dougherty, E. Salturk and S. Szabo, On codes over local rings: generator matrices, generating characters and MacWilliams identities, submitted. Google Scholar

[9]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available at http://codetables.de. Google Scholar

[10]

A. R. Hammons Jr.P. V. KumarA. R. CalderbankN. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdoc, Preparata, Goethals and related code, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[11]

T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 3 (2006), 319-335.   Google Scholar

[12]

P. Kanwar and S. Lopez-Permouth, Cyclic codes over the integers modulo pm, Finite Fields Appl., 3 (1997), 334-352.  doi: 10.1006/ffta.1997.0189.  Google Scholar

[13]

E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Contemp. Math. , to appear. doi: 10.1090/conm/634/12702.  Google Scholar

[14]

B. R. McDonald, Finite Rings with Identity, Dekker, New York, 1974.  Google Scholar

[15]

V. PlessP. Solé and Z. Qian, Cyclic self-dual $\mathbb Z_4$-codes, Finite Fields Appl., 3 (1997), 48-69.  doi: 10.1006/ffta.1996.0172.  Google Scholar

[16]

E. Prange, Cyclic Error-Correcting Codes in Two Symbols, Air Force Cambridge Research Center 1957. Google Scholar

[17]

J. Wolfmann, Negacyclic and cyclic codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, 45 (1999), 2527-2532.  doi: 10.1109/18.796397.  Google Scholar

[18]

B. Yildiz and S. Karadeniz, Cyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2$, Des. Codes Cryptogr., 58 (2011), 221-234.  doi: 10.1007/s10623-010-9399-3.  Google Scholar

show all references

References:
[1]

N. Aydin and T. Asamov, A database of $\mathbb Z_4$-codes, available at http://www.asamov.com/Z4Codes. Google Scholar

[2]

A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb{F}_2 + u\mathbb{F}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1109/18.761278.  Google Scholar

[3]

A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptpgr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.  Google Scholar

[4]

E. Z. Chen, A database of quasi-twisted codes, available at http://moodle.tec.hkr.se/~chen/research/codes. Google Scholar

[5]

H. Q. Dinh and S. Lopez-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

[6]

S. T. DoughertyS. Karadeniz and B. Yildiz, Cyclic codes over Rk, Des. Codes Cryptogr., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.  Google Scholar

[7]

S. T. DoughertyE. Salturk and S. Szabo, Codes over local rings of order 16 and binary codes, Adv. math. Comm., 10 (), 379-391.  doi: 10.3934/amc.2016012.  Google Scholar

[8]

S. T. Dougherty, E. Salturk and S. Szabo, On codes over local rings: generator matrices, generating characters and MacWilliams identities, submitted. Google Scholar

[9]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available at http://codetables.de. Google Scholar

[10]

A. R. Hammons Jr.P. V. KumarA. R. CalderbankN. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdoc, Preparata, Goethals and related code, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[11]

T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 3 (2006), 319-335.   Google Scholar

[12]

P. Kanwar and S. Lopez-Permouth, Cyclic codes over the integers modulo pm, Finite Fields Appl., 3 (1997), 334-352.  doi: 10.1006/ffta.1997.0189.  Google Scholar

[13]

E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Contemp. Math. , to appear. doi: 10.1090/conm/634/12702.  Google Scholar

[14]

B. R. McDonald, Finite Rings with Identity, Dekker, New York, 1974.  Google Scholar

[15]

V. PlessP. Solé and Z. Qian, Cyclic self-dual $\mathbb Z_4$-codes, Finite Fields Appl., 3 (1997), 48-69.  doi: 10.1006/ffta.1996.0172.  Google Scholar

[16]

E. Prange, Cyclic Error-Correcting Codes in Two Symbols, Air Force Cambridge Research Center 1957. Google Scholar

[17]

J. Wolfmann, Negacyclic and cyclic codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, 45 (1999), 2527-2532.  doi: 10.1109/18.796397.  Google Scholar

[18]

B. Yildiz and S. Karadeniz, Cyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2$, Des. Codes Cryptogr., 58 (2011), 221-234.  doi: 10.1007/s10623-010-9399-3.  Google Scholar

Table 1.  One-generator cyclic codes over $\mathbb{F}_{2}\left[u, v\right] /\left\langle u^{2}+v^{2}, uv\right\rangle $
$n$ $f\left(x\right) $ $\phi \left({C}\right) $
3 $\left(0, z_{4}, z_{4}\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
3 $\left(z_{7}, z_{2}, z_{8}\right) $ $\left[12, 4, 6\right] _{2}^{\ast }$
3 $\left(z_{1}, z_{8}, z_{6}\right) $ $\left[12, 6, 4\right] _{2}^{\ast }$
3 $\left(y_{7}, y_{5}, y_{2}\right) $ $\left[12, 8, 3\right] _{2}^{\ast }$
5 $\left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right) $ $\left[20, 8, 8\right] _{2}^{\ast }$
5 $\left(y_{4}, y_{1}, y_{3}, y_{2}, y_{5}\right) $ $\left[20, 12, 4\right] _{2}^{\ast }$
7 $\left(z_{4}, z_{1}, z_{4}, z_{1}, z_{1}, z_{4}, z_{4}\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
7 $\left(z_{7}, z_{5}, z_{7}, z_{6}, z_{6}, z_{5}, z_{1}\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
7 $\left(z_{6}, z_{5}, z_{6}, z_{3}, z_{3}, z_{5}, z_{4}\right) $ $\left[28, 7, 12\right] _{2}^{\ast }$
7 $\left(z_{7}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 8, 10\right] _{2}^{\ast -1}$
7 $\left(z_{5}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
7 $\left(y_{1}, y_{1}, y_{4}, z_{2}, y_{6}, z_{8}, z_{5}\right) $ $\left[28, 19, 4\right] _{2}^{\ast }$
7 $\left(y_{2}, y_{4}, y_{4}, z_{8}, y_{1}, z_{5}, z_{1}\right) $ $\left[28, 20, 4\right] _{2}^{\ast }$
9 $\left(z_{8}, z_{8}, z_{3}, z_{4}, z_{3}, z_{6}, z_{2}, z_{5}, z_{5}\right) $ $\left[36, 16, 8\right] _{2}^{\ast -2}$
9 $\left(y_{7}, z_{6}, y_{2}, y_{5}, z_{1}, y_{8}, y_{2}, z_{1}, y_{8}\right) $ $\left[36, 20, 6\right] _{2}^{\ast -2}$
15 $\left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right) $ $\left[60, 20, 16\right] _{2}^{b-1}$
15 $\left(z_{4}, y_{7}, z_{7}, z_{1}, y_{5}, z_{5}, z_{3}, y_{8}, z_{8}, y_{4}, z_{4}, z_{2}, z_{3}, z_{6}, z_{7}\right) $ $\left[60, 50, 4\right] _{2}^{\ast }$
17 $\left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right) $ $\left[68, 48, 6\right] _{2}^{\ast -2}$
21 $\left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right) $ $\left[84, 58, 8\right] _{2}^{b}$
21 $\left(y_{2}y_{6}y_{4}y_{7}y_{5}z_{8}z_{3}y_{1}z_{5}y_{8}y_{4}z_{4}y_{3}z_{2}z_{3}z_{7}z_{4}z_{6}z_{8}y_{3}y_{8}\right) $ $\left[84, 62, 6\right] _{2}^{b-2}$
21 $\left(y_{2}z_{2}z_{6}y_{8}z_{3}y_{3}y_{3}z_{3}z_{1}z_{6}z_{8}z_{2}y_{1}z_{8}z_{7}z_{2}y_{1}y_{4}z_{8}y_{5}z_{7}\right) $ $\left[84, 72, 4\right] _{2}^{b}$
23 $% (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}) $ $\left[92, 66, 8\right] _{2}^{\ast }$
$n$ $f\left(x\right) $ $\phi \left({C}\right) $
3 $\left(0, z_{4}, z_{4}\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
3 $\left(z_{7}, z_{2}, z_{8}\right) $ $\left[12, 4, 6\right] _{2}^{\ast }$
3 $\left(z_{1}, z_{8}, z_{6}\right) $ $\left[12, 6, 4\right] _{2}^{\ast }$
3 $\left(y_{7}, y_{5}, y_{2}\right) $ $\left[12, 8, 3\right] _{2}^{\ast }$
5 $\left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right) $ $\left[20, 8, 8\right] _{2}^{\ast }$
5 $\left(y_{4}, y_{1}, y_{3}, y_{2}, y_{5}\right) $ $\left[20, 12, 4\right] _{2}^{\ast }$
7 $\left(z_{4}, z_{1}, z_{4}, z_{1}, z_{1}, z_{4}, z_{4}\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
7 $\left(z_{7}, z_{5}, z_{7}, z_{6}, z_{6}, z_{5}, z_{1}\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
7 $\left(z_{6}, z_{5}, z_{6}, z_{3}, z_{3}, z_{5}, z_{4}\right) $ $\left[28, 7, 12\right] _{2}^{\ast }$
7 $\left(z_{7}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 8, 10\right] _{2}^{\ast -1}$
7 $\left(z_{5}, z_{2}, z_{2}, z_{4}, z_{5}, z_{7}, z_{4}\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
7 $\left(y_{1}, y_{1}, y_{4}, z_{2}, y_{6}, z_{8}, z_{5}\right) $ $\left[28, 19, 4\right] _{2}^{\ast }$
7 $\left(y_{2}, y_{4}, y_{4}, z_{8}, y_{1}, z_{5}, z_{1}\right) $ $\left[28, 20, 4\right] _{2}^{\ast }$
9 $\left(z_{8}, z_{8}, z_{3}, z_{4}, z_{3}, z_{6}, z_{2}, z_{5}, z_{5}\right) $ $\left[36, 16, 8\right] _{2}^{\ast -2}$
9 $\left(y_{7}, z_{6}, y_{2}, y_{5}, z_{1}, y_{8}, y_{2}, z_{1}, y_{8}\right) $ $\left[36, 20, 6\right] _{2}^{\ast -2}$
15 $\left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right) $ $\left[60, 20, 16\right] _{2}^{b-1}$
15 $\left(z_{4}, y_{7}, z_{7}, z_{1}, y_{5}, z_{5}, z_{3}, y_{8}, z_{8}, y_{4}, z_{4}, z_{2}, z_{3}, z_{6}, z_{7}\right) $ $\left[60, 50, 4\right] _{2}^{\ast }$
17 $\left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right) $ $\left[68, 48, 6\right] _{2}^{\ast -2}$
21 $\left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right) $ $\left[84, 58, 8\right] _{2}^{b}$
21 $\left(y_{2}y_{6}y_{4}y_{7}y_{5}z_{8}z_{3}y_{1}z_{5}y_{8}y_{4}z_{4}y_{3}z_{2}z_{3}z_{7}z_{4}z_{6}z_{8}y_{3}y_{8}\right) $ $\left[84, 62, 6\right] _{2}^{b-2}$
21 $\left(y_{2}z_{2}z_{6}y_{8}z_{3}y_{3}y_{3}z_{3}z_{1}z_{6}z_{8}z_{2}y_{1}z_{8}z_{7}z_{2}y_{1}y_{4}z_{8}y_{5}z_{7}\right) $ $\left[84, 72, 4\right] _{2}^{b}$
23 $% (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}) $ $\left[92, 66, 8\right] _{2}^{\ast }$
Table 2.  Two-generator cyclic codes over $\mathbb{F}_{2}\left[u, v\right] /\left\langle u^{2}+v^{2}, uv\right\rangle $
$n$ $\left. f\left(x\right), g\left(x\right) \right. $ $\phi \left({C}\right) $
3 $\left. \left(z_{6}, z_{5, }z_{7}\right), \left(z_{8}, z_{8}, z_{8}\right) \right. $ $\left[12, 6, 4\right] _{2}^{\ast }$
3 $\left. \left(z_{8}, z_{6}, z_{7}\right), \left(z_{5}, z_{3}, z_{6}\right) \right. $ $\left[12, 7, 4\right] _{2}^{\ast }$
5 $\left. \left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right), \left(z_{3}, z_{3}, z_{3}, z_{7}, z_{1}\right) \right. $ $\left[20, 13, 4\right] _{2}^{\ast }$
7 $\left. \left(z_{7}, z_{4}, z_{4}, z_{6}, z_{2}, z_{4}, z_{7}\right), \left(z_{8}, z_{5}, z_{1}, z_{1}, z_{4}, z_{2}, z_{2}\right) \right. $ $\left[28, 15, 6% \right] _{2}^{\ast }$
7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{5}, y_{2}z_{8}, y_{5}, y_{1}, y_{1}, z_{7}\right) \right. $ $\left[28, 21, 4% \right] _{2}^{\ast }$
7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{4}, z_{3}\right) \right. $ $\left[28, 22, 4% \right] _{2}^{\ast }$
15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{2}, z_{7}, z_{3}, z_{7}, z_{3}, z_{1}, z_{8}, z_{6}, z_{3}, z_{2}, z_{1}, z_{4}, z_{4}, z_{3}, z_{6}\right)% \end{array}% \right. $ $\left[60, 28, 10\right] _{2}^{b-2}$
15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{4}, y_{1}, z_{3}, z_{7}, z_{8}, y_{2}, z_{4}, z_{5}, y_{2}, y_{5}, y_{5}, y_{7}, y_{8}, y_{6}, z_{4}\right)% \end{array}% \right.$ $\left[60, 44, 6\right] _{2}^{b}$
17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(y_{5}, y_{8}, z_{2}, y_{7}, y_{5}, z_{2}, z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{2}, y_{8}, z_{4}, y_{6}, z_{6}, z_{2}\right)% \end{array}% \right.$ $\left[68, 56, 4\right] _{2}^{b}$
17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(z_{1}, z_{5}, y_{3}, y_{8}, z_{6}, z_{4}, z_{2}, z_{6}, y_{7}, y_{5}, z_{3}, z_{2}, y_{2}, z_{3}, z_{7}, z_{6}, y_{2}\right)% \end{array}% \right.$ $\left[68, 57, 4\right] _{2}^{b}$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{3}z_{6}z_{3}z_{1}z_{1}z_{8}z_{3}z_{1}z_{6}z_{6}z_{4}z_{2}z_{3}z_{5}z_{8}z_{4}z_{5}z_{4}z_{7}z_{1}z_{7}\right)% \end{array}% \right.$ $\left[84, 63, 6\right] _{2}^{\ast -2}$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{6}y_{4}y_{8}z_{3}z_{7}z_{3}y_{5}y_{7}y_{5}z_{7}y_{7}z_{8}y_{4}z_{8}y_{3}y_{2}y_{4}z_{5}y_{3}z_{1}y_{2}\right)% \end{array}% \right.$ $\left[84, 75, 4\right] _{2}^{\ast }$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(y_{6}y_{5}z_{7}y_{5}y_{5}z_{1}y_{2}y_{4}y_{6}z_{6}z_{4}y_{3}z_{8}z_{1}z_{5}y_{4}y_{4}z_{7}z_{8}z_{8}z_{6}\right)% \end{array}% \right.$ $\left[84, 76, 4\right] _{2}^{\ast }$
23 $\left. \begin{array}{c} (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}), \\ \left(z_{5}z_{3}z_{1}z_{7}z_{6}z_{3}z_{5}z_{1}z_{3}z_{2}z_{5}z_{2}z_{2}z_{4}z_{7}z_{1}z_{1}z_{3}z_{2}z_{8}z_{3}z_{8}z_{4}\right)% \end{array}% \right. $ $\left[92, 78, 4\right] _{2}^{b-1}$
$n$ $\left. f\left(x\right), g\left(x\right) \right. $ $\phi \left({C}\right) $
3 $\left. \left(z_{6}, z_{5, }z_{7}\right), \left(z_{8}, z_{8}, z_{8}\right) \right. $ $\left[12, 6, 4\right] _{2}^{\ast }$
3 $\left. \left(z_{8}, z_{6}, z_{7}\right), \left(z_{5}, z_{3}, z_{6}\right) \right. $ $\left[12, 7, 4\right] _{2}^{\ast }$
5 $\left. \left(z_{7}, z_{2}, z_{2}, z_{7}, z_{1}\right), \left(z_{3}, z_{3}, z_{3}, z_{7}, z_{1}\right) \right. $ $\left[20, 13, 4\right] _{2}^{\ast }$
7 $\left. \left(z_{7}, z_{4}, z_{4}, z_{6}, z_{2}, z_{4}, z_{7}\right), \left(z_{8}, z_{5}, z_{1}, z_{1}, z_{4}, z_{2}, z_{2}\right) \right. $ $\left[28, 15, 6% \right] _{2}^{\ast }$
7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{5}, y_{2}z_{8}, y_{5}, y_{1}, y_{1}, z_{7}\right) \right. $ $\left[28, 21, 4% \right] _{2}^{\ast }$
7 $\left. \left(z_{5}, z_{5}, z_{7}, z_{2}, z_{4}, z_{7}, z_{6}\right), \left(z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{4}, z_{3}\right) \right. $ $\left[28, 22, 4% \right] _{2}^{\ast }$
15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{2}, z_{7}, z_{3}, z_{7}, z_{3}, z_{1}, z_{8}, z_{6}, z_{3}, z_{2}, z_{1}, z_{4}, z_{4}, z_{3}, z_{6}\right)% \end{array}% \right. $ $\left[60, 28, 10\right] _{2}^{b-2}$
15 $\left. \begin{array}{c} \left(z_{5}, z_{3}, z_{4}, z_{1}, z_{4}, z_{6}, z_{7}, z_{5}, z_{7}, z_{1}, z_{3}, z_{6}, z_{3}, z_{4}, z_{7}\right), \\ \left(z_{4}, y_{1}, z_{3}, z_{7}, z_{8}, y_{2}, z_{4}, z_{5}, y_{2}, y_{5}, y_{5}, y_{7}, y_{8}, y_{6}, z_{4}\right)% \end{array}% \right.$ $\left[60, 44, 6\right] _{2}^{b}$
17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(y_{5}, y_{8}, z_{2}, y_{7}, y_{5}, z_{2}, z_{7}, y_{5}, z_{1}, y_{7}, y_{2}, y_{2}, y_{8}, z_{4}, y_{6}, z_{6}, z_{2}\right)% \end{array}% \right.$ $\left[68, 56, 4\right] _{2}^{b}$
17 $\left. \begin{array}{c} \left(z_{7}, z_{1}, z_{2}, z_{8}, y_{2}, y_{7}, y_{7}, z_{1}, z_{3}, y_{2}, y_{3}, z_{6}, z_{3}, y_{4}, y_{4}, y_{3}, z_{6}\right), \\ \left(z_{1}, z_{5}, y_{3}, y_{8}, z_{6}, z_{4}, z_{2}, z_{6}, y_{7}, y_{5}, z_{3}, z_{2}, y_{2}, z_{3}, z_{7}, z_{6}, y_{2}\right)% \end{array}% \right.$ $\left[68, 57, 4\right] _{2}^{b}$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{3}z_{6}z_{3}z_{1}z_{1}z_{8}z_{3}z_{1}z_{6}z_{6}z_{4}z_{2}z_{3}z_{5}z_{8}z_{4}z_{5}z_{4}z_{7}z_{1}z_{7}\right)% \end{array}% \right.$ $\left[84, 63, 6\right] _{2}^{\ast -2}$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(z_{6}y_{4}y_{8}z_{3}z_{7}z_{3}y_{5}y_{7}y_{5}z_{7}y_{7}z_{8}y_{4}z_{8}y_{3}y_{2}y_{4}z_{5}y_{3}z_{1}y_{2}\right)% \end{array}% \right.$ $\left[84, 75, 4\right] _{2}^{\ast }$
21 $\left. \begin{array}{c} \left(z_{4}y_{3}y_{1}y_{8}z_{3}y_{2}z_{3}z_{2}z_{2}y_{2}y_{7}y_{2}y_{5}y_{8}z_{1}z_{2}y_{5}y_{6}y_{1}z_{6}z_{1}\right), \\ \left(y_{6}y_{5}z_{7}y_{5}y_{5}z_{1}y_{2}y_{4}y_{6}z_{6}z_{4}y_{3}z_{8}z_{1}z_{5}y_{4}y_{4}z_{7}z_{8}z_{8}z_{6}\right)% \end{array}% \right.$ $\left[84, 76, 4\right] _{2}^{\ast }$
23 $\left. \begin{array}{c} (y_{4}y_{4}y_{4}y_{1}y_{6}z_{8}y_{1}z_{6}y_{2}y_{2}z_{1}z_{4}y_{4}y_{1}z_{4}z_{6}y_{1}z_{7}y_{6}z_{1}z_{7}z_{3}z_{6}), \\ \left(z_{5}z_{3}z_{1}z_{7}z_{6}z_{3}z_{5}z_{1}z_{3}z_{2}z_{5}z_{2}z_{2}z_{4}z_{7}z_{1}z_{1}z_{3}z_{2}z_{8}z_{3}z_{8}z_{4}\right)% \end{array}% \right. $ $\left[92, 78, 4\right] _{2}^{b-1}$
Table 3.  Some cyclic codes over $\mathbb{Z}_{4}\left[u\right] /\left\langle u^{2}-2\right\rangle $
$n$ $f\left(x\right) $ $\phi _{\mathbb{Z}_{4}}\left({C}\right) $ $\phi \left({C}\right) $
3 $\left(00|02|02\right) $ $\left(3, 4^{0}2^{2}, 8\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
3 $\left(03|03|03\right) $ $\left(3, 4^{1}2^{1}, 6\right) $ $\left[12, 3, 6\right] _{2}^{\ast }$
3 $\left(01|00|01\right) $ $\left(3, 4^{2}2^{3}, 6\right) $ $\left[12, 7, 4\right] _{2}^{\ast }$
5 $\left(21|21|21|21|21\right) $ $\left(10, 4^{1}2^{1}, 10\right) $ $% \left[20, 3, 10\right] _{2}^{\ast -1}$
5 $\left(01|22|01|01|23\right) $ $\left(10, 4^{4}2^{4}, 4\right) $ $% \left(20, 2^{12}, 4\right) _{2}^{\ast }$
5 $\left(03|03|21|23|00\right) $ $\left(10, 4^{4}2^{5}, 4\right) $ $% \left[20, 13, 4\right] _{2}^{\ast }$
7 $\left(00|02|00|02|02|02|00\right) $ $\left(14, 4^{0}2^{3}, 8\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
7 $\left(23|21|21|23|23|23|21\right) $ $\left(14, 4^{1}2^{4}, 8\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
7 $\left(22|23|00|01|23|01|22\right) $ $\left(14, 4^{3}2^{6}, 8\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
7 $\left(11|13|23|20|02|33|00\right) $ $\left(14, 4^{6}2^{3}, 6\right) $ $\left(28, 2^{15}, 6\right) _{2}^{\ast }$
7 $\left(23|00|30|01|11|33|02\right) $ $\left(14, 4^{8}2^{3}, 4\right) $ $\left(28, 2^{19}, 4\right) _{2}^{\ast }$
7 $\left(33|30|11|22|12|00|20\right) $ $\left(14, 4^{9}2^{3}, 4\right) $ $\left(28, 2^{21}, 4\right) _{2}^{\ast }$
7 $\left(32|01|33|33|11|21|03\right) $ $\left(14, 4^{9}2^{4}, 4\right) $ $\left(28, 2^{22}, 4\right) _{2}^{\ast }$
9 $\left(33|23|33|32|03|30|13|21|32\right) $ $\left(18, 4^{10}2^{6}, 4\right) $ $\left(36, 2^{26}, 4\right) _{2}^{\ast }$
9 $\left(12|02|11|32|21|10|30|20|32\right) $ $\left(18, 4^{10}2^{7}, 4\right) $ $\left(36, 2^{27}, 4\right) _{2}^{\ast }$
17 $\left(02|20|01|30|32|32|02|03|31|31|32|21|33|20|21\right) $ $\left(34, 4^{24}2^{9}, 4\right) $ $\left(68, 2^{57}, 4\right) _{2}^{\ast }$
$n$ $f\left(x\right) $ $\phi _{\mathbb{Z}_{4}}\left({C}\right) $ $\phi \left({C}\right) $
3 $\left(00|02|02\right) $ $\left(3, 4^{0}2^{2}, 8\right) $ $\left[12, 2, 8\right] _{2}^{\ast }$
3 $\left(03|03|03\right) $ $\left(3, 4^{1}2^{1}, 6\right) $ $\left[12, 3, 6\right] _{2}^{\ast }$
3 $\left(01|00|01\right) $ $\left(3, 4^{2}2^{3}, 6\right) $ $\left[12, 7, 4\right] _{2}^{\ast }$
5 $\left(21|21|21|21|21\right) $ $\left(10, 4^{1}2^{1}, 10\right) $ $% \left[20, 3, 10\right] _{2}^{\ast -1}$
5 $\left(01|22|01|01|23\right) $ $\left(10, 4^{4}2^{4}, 4\right) $ $% \left(20, 2^{12}, 4\right) _{2}^{\ast }$
5 $\left(03|03|21|23|00\right) $ $\left(10, 4^{4}2^{5}, 4\right) $ $% \left[20, 13, 4\right] _{2}^{\ast }$
7 $\left(00|02|00|02|02|02|00\right) $ $\left(14, 4^{0}2^{3}, 8\right) $ $\left[28, 3, 16\right] _{2}^{\ast }$
7 $\left(23|21|21|23|23|23|21\right) $ $\left(14, 4^{1}2^{4}, 8\right) $ $\left[28, 6, 12\right] _{2}^{\ast }$
7 $\left(22|23|00|01|23|01|22\right) $ $\left(14, 4^{3}2^{6}, 8\right) $ $\left[28, 12, 8\right] _{2}^{\ast }$
7 $\left(11|13|23|20|02|33|00\right) $ $\left(14, 4^{6}2^{3}, 6\right) $ $\left(28, 2^{15}, 6\right) _{2}^{\ast }$
7 $\left(23|00|30|01|11|33|02\right) $ $\left(14, 4^{8}2^{3}, 4\right) $ $\left(28, 2^{19}, 4\right) _{2}^{\ast }$
7 $\left(33|30|11|22|12|00|20\right) $ $\left(14, 4^{9}2^{3}, 4\right) $ $\left(28, 2^{21}, 4\right) _{2}^{\ast }$
7 $\left(32|01|33|33|11|21|03\right) $ $\left(14, 4^{9}2^{4}, 4\right) $ $\left(28, 2^{22}, 4\right) _{2}^{\ast }$
9 $\left(33|23|33|32|03|30|13|21|32\right) $ $\left(18, 4^{10}2^{6}, 4\right) $ $\left(36, 2^{26}, 4\right) _{2}^{\ast }$
9 $\left(12|02|11|32|21|10|30|20|32\right) $ $\left(18, 4^{10}2^{7}, 4\right) $ $\left(36, 2^{27}, 4\right) _{2}^{\ast }$
17 $\left(02|20|01|30|32|32|02|03|31|31|32|21|33|20|21\right) $ $\left(34, 4^{24}2^{9}, 4\right) $ $\left(68, 2^{57}, 4\right) _{2}^{\ast }$
Table 4.  Some cyclic codes over $\mathbb{Z}_{16}$
$n$ $f\left(x\right) $ ${C}$ $\phi \left({C}\right) $
3 $\left(8, 8, 0\right) $ $\left[3, 2^{2}, 8\right] $ $\left(12, 2^{2}, 8\right) _{2}^{\ast }$
3 $\left(6, 6, 6\right) $ $\left[3, 8^{1}, 6\right] $ $\left(12, 2^{3}, 6\right) _{2}^{\ast }$
3 $\left(12, 4, 8\right) $ $\left[3, 4^{2}2^{1}, 4\right] $ $\left(12, 2^{5}, 4\right) _{2}^{\ast }$
3 $\left(4, 6, 6\right) $ $\left[3, 8^{2}, 4\right] $ $\left(12, 2^{6}, 4\right) _{2}^{\ast }$
3 $\left(2, 2, 4\right) $ $\left[3, 8^{2}2^{1}, 4\right] $ $\left(12, 2^{7}, 4\right) _{2}^{\ast }$
3 $\left(11, 3, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(12, 2^{11}, 2\right) _{2}^{\ast }$
5 $\left(14, 10, 6, 6, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(20, 2^{12}, 4\right) _{2}^{\ast }$
5 $\left(12, 14, 8, 4, 2\right) $ $\left[5, 8^{4}2^{1}, 4\right] $ $\left(20, 2^{13}, 4\right) _{2}^{\ast }$
5 $\left(13, 6, 7, 10, 12\right) $ $\left[5, 16^{4}, 2\right] $ $\left(20, 2^{16}, 2\right) _{2}^{\ast }$
5 $\left(7, 5, 3, 11, 14\right) $ $\left[5, 16^{4}2^{1}, 2\right] $ $% \left(20, 2^{17}, 2\right) _{2}^{\ast }$
7 $\left(10, 12, 6, 10, 2, 8, 0\right) $ $\left[7, 8^{3}, 10\right] $ $% \left(28, 2^{9}, 10\right) _{2}^{\ast }$
7 $\left(2, 12, 12, 6, 12, 14, 6\right) $ $\left[7, 8^{3}2^{3}, 8\right] $ $% \left(28, 2^{12}, 8\right) _{2}^{\ast }$
7 $\left(13, 11, 7, 13, 1, 5, 7\right) $ $\left[7, 16^{1}8^{3}, 7\right] $ $% \left(28, 2^{13}, 7\right) _{2}^{\ast -1}$
7 $\left(1, 4, 14, 7, 11, 7, 4\right) $ $\left[7, 16^{3}2^{3}, 6\right] $ $% \left(28, 2^{15}, 6\right) _{2}^{\ast }$
$n$ $f\left(x\right) $ ${C}$ $\phi \left({C}\right) $
3 $\left(8, 8, 0\right) $ $\left[3, 2^{2}, 8\right] $ $\left(12, 2^{2}, 8\right) _{2}^{\ast }$
3 $\left(6, 6, 6\right) $ $\left[3, 8^{1}, 6\right] $ $\left(12, 2^{3}, 6\right) _{2}^{\ast }$
3 $\left(12, 4, 8\right) $ $\left[3, 4^{2}2^{1}, 4\right] $ $\left(12, 2^{5}, 4\right) _{2}^{\ast }$
3 $\left(4, 6, 6\right) $ $\left[3, 8^{2}, 4\right] $ $\left(12, 2^{6}, 4\right) _{2}^{\ast }$
3 $\left(2, 2, 4\right) $ $\left[3, 8^{2}2^{1}, 4\right] $ $\left(12, 2^{7}, 4\right) _{2}^{\ast }$
3 $\left(11, 3, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(12, 2^{11}, 2\right) _{2}^{\ast }$
5 $\left(14, 10, 6, 6, 12\right) $ $\left[5, 8^{4}, 4\right] $ $\left(20, 2^{12}, 4\right) _{2}^{\ast }$
5 $\left(12, 14, 8, 4, 2\right) $ $\left[5, 8^{4}2^{1}, 4\right] $ $\left(20, 2^{13}, 4\right) _{2}^{\ast }$
5 $\left(13, 6, 7, 10, 12\right) $ $\left[5, 16^{4}, 2\right] $ $\left(20, 2^{16}, 2\right) _{2}^{\ast }$
5 $\left(7, 5, 3, 11, 14\right) $ $\left[5, 16^{4}2^{1}, 2\right] $ $% \left(20, 2^{17}, 2\right) _{2}^{\ast }$
7 $\left(10, 12, 6, 10, 2, 8, 0\right) $ $\left[7, 8^{3}, 10\right] $ $% \left(28, 2^{9}, 10\right) _{2}^{\ast }$
7 $\left(2, 12, 12, 6, 12, 14, 6\right) $ $\left[7, 8^{3}2^{3}, 8\right] $ $% \left(28, 2^{12}, 8\right) _{2}^{\ast }$
7 $\left(13, 11, 7, 13, 1, 5, 7\right) $ $\left[7, 16^{1}8^{3}, 7\right] $ $% \left(28, 2^{13}, 7\right) _{2}^{\ast -1}$
7 $\left(1, 4, 14, 7, 11, 7, 4\right) $ $\left[7, 16^{3}2^{3}, 6\right] $ $% \left(28, 2^{15}, 6\right) _{2}^{\ast }$
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