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February  2020, 14(1): 53-67. doi: 10.3934/amc.2020005

Skew constacyclic codes over the local Frobenius non-chain rings of order 16

Nuh Aydin 1, , Yasemin Cengellenmis 2, , Abdullah Dertli 3, , Steven T. Dougherty 4,, and  Esengül Saltürk 5,

1. 

Department of Mathematics, Kenyon College, Gambier, OH 43022, USA
2. 

Department of Mathematics, Trakya University, 22030 Edirne, Turkey
3. 

Department of Mathematics, Ondokuz Mayis University, 55139 Samsun, Turkey
4. 

Department of Mathematics, University of Scranton, Scranton, PA. 18518, USA
5. 

The Scientific and Technological, Research Council Of Turkey, 41401 Kocaeli, Turkey

* Corresponding author: Steven T. Dougherty

Esengül Saltürk would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper

Received  June 2018 Revised  December 2018 Published  August 2019

We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

Keywords: Skew-cyclic code, Frobenius rings, local rings.
Mathematics Subject Classification: Primary: 11T71; Secondary: 94B15, 13H99.
Citation: Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005
References:
[1]

N. Aydin and T. Asamov, A Database of $\mathbb{Z}_4$ Codes, Journal of Combinatorics, Information & System Sciences, 34 (2009), 1-12. 

[2]

N. Aydin, A. Dertli and Y. Cengellenmis, Cyclic and constacyclic codes over $\mathbb{Z}_4+w\mathbb{Z}_4$, preprint.

[3]

N. Aydin, I. Siap and D. K. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Cryptogr., 24 (2001), 313-326.  doi: 10.1023/A:1011283523000.

[4]

V. K. Bhargava, G. E. Séguin and J. M. Stein, Some $(ink, k)$ cyclic codes in quasi-cyclic form, IEEE Trans. Inform. Theory, 24 (1978), 630-632.  doi: 10.1109/TIT.1978.1055930.

[5]

T. Blackford, Cyclic codes over $Z_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.

[6]

I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[7]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, App. Algebra in Eng. Comm. and Comp., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[8]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over galois rings, Advances in Mathematics of Communications, 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[9]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Lecture Notes in Computer Science, 5921 (2009), 38-55.  doi: 10.1007/978-3-642-10868-6_3.

[10]

C. L. Chen, W. W. Peterson and E. J. Weldon, Some results on quasi-cyclic codes, Information and Control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.

[11]

S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, Spinger-Verlag, 2017. doi: 10.1007/978-3-319-59806-2.

[12]

S. T. Dougherty, A. Kaya and E. Saltürk, Cyclic codes over local rings of order $16$, Adv. Math. Commun., 11 (2017), 99-114.  doi: 10.3934/amc.2017005.

[13]

S. T. Dougherty and E. Saltürk, Codes over a family of local Frobenius rings, Gray maps and self-dual codes, Discrete Appl. Math., 217 (2017), 512-524.  doi: 10.1016/j.dam.2016.09.025.

[14]

S. T. Dougherty and E. Salturk, Constacyclic codes over local rings of order $16$, 2017 (in submission).

[15]

S. T. Dougherty, E. Saltürk and S. Szabo, On codes over local rings: Generator matrices, generating characters and MacWilliams identities, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 193-206. 

[16]

S. T. Dougherty, E. Saltürk and S. Szabo, Codes over local rings of order 16 and binary codes, Adv. Math. Commun., 10 (2016), 379-391.  doi: 10.3934/amc.2016012.

[17]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.

[18]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Cyclic Codes over $R_k$, Des. Codes Cryptogr., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.

[19]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-Dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106. 

[20]

M. Greferath, Cyclic codes over finite rings, Discrete Mathematics, 177 (1997), 273-277.  doi: 10.1016/S0012-365X(97)00006-X.

[21]

T. A. Gulliver, Construction Of Quasi-Cyclic Codes, Ph. D. Dissertation, University of New Brunswick, 1984.

[22]

T. A. Gulliver and V. K. Bhargava, A (105, 10, 47) binary quasi-cyclic code, App. Math. Lett., 8 (1995), 67-70.  doi: 10.1016/0893-9659(95)00049-V.

[23]

S. Ling, H. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Des. Codes Cryptogr., 38 (2006), 337-361.  doi: 10.1007/s10623-005-1431-7.

[24]

S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅱ: Chain rings, Des. Codes Cryptogr., 30 (2003), 113-130.  doi: 10.1023/A:1024715527805.

[25]

E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Noncommutative rings and their applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 634 (2015), 227-241. doi: 10.1090/conm/634/12702.

[26]

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

[27]

J. Wolfmann, Negacyclic and cyclic codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 45 (1999), 2527-2532.  doi: 10.1109/18.796397.

[28]

J. Wood, Lecture Notes On Dual Codes And the MacWilliams Identities, Mexico, 2009.

[29]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, The American Journal of Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.

[30]

Magma computer algebra system, online, http://magma.maths.usyd.edu.au/.

[31]

, A Database on Binary Quasi-Cyclic Codes, online, Accessed January, 2018, http://www.tec.hkr.se/ chen/research/codes/qc.htm.

[32]

Code tables: Bounds on the parameters of codes, online, Accessed January, 2018, http://www.codetables.de/.

[33]

Database of $\mathbb{Z}_4$ codes, online, $Z_4$Codes.info, Accessed February, 2017.

show all references

References:
[1]

N. Aydin and T. Asamov, A Database of $\mathbb{Z}_4$ Codes, Journal of Combinatorics, Information & System Sciences, 34 (2009), 1-12. 

[2]

N. Aydin, A. Dertli and Y. Cengellenmis, Cyclic and constacyclic codes over $\mathbb{Z}_4+w\mathbb{Z}_4$, preprint.

[3]

N. Aydin, I. Siap and D. K. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Cryptogr., 24 (2001), 313-326.  doi: 10.1023/A:1011283523000.

[4]

V. K. Bhargava, G. E. Séguin and J. M. Stein, Some $(ink, k)$ cyclic codes in quasi-cyclic form, IEEE Trans. Inform. Theory, 24 (1978), 630-632.  doi: 10.1109/TIT.1978.1055930.

[5]

T. Blackford, Cyclic codes over $Z_4$ of oddly even length, Discrete Applied Mathematics, 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.

[6]

I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[7]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, App. Algebra in Eng. Comm. and Comp., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[8]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over galois rings, Advances in Mathematics of Communications, 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[9]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Lecture Notes in Computer Science, 5921 (2009), 38-55.  doi: 10.1007/978-3-642-10868-6_3.

[10]

C. L. Chen, W. W. Peterson and E. J. Weldon, Some results on quasi-cyclic codes, Information and Control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.

[11]

S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, Spinger-Verlag, 2017. doi: 10.1007/978-3-319-59806-2.

[12]

S. T. Dougherty, A. Kaya and E. Saltürk, Cyclic codes over local rings of order $16$, Adv. Math. Commun., 11 (2017), 99-114.  doi: 10.3934/amc.2017005.

[13]

S. T. Dougherty and E. Saltürk, Codes over a family of local Frobenius rings, Gray maps and self-dual codes, Discrete Appl. Math., 217 (2017), 512-524.  doi: 10.1016/j.dam.2016.09.025.

[14]

S. T. Dougherty and E. Salturk, Constacyclic codes over local rings of order $16$, 2017 (in submission).

[15]

S. T. Dougherty, E. Saltürk and S. Szabo, On codes over local rings: Generator matrices, generating characters and MacWilliams identities, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 193-206. 

[16]

S. T. Dougherty, E. Saltürk and S. Szabo, Codes over local rings of order 16 and binary codes, Adv. Math. Commun., 10 (2016), 379-391.  doi: 10.3934/amc.2016012.

[17]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.

[18]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Cyclic Codes over $R_k$, Des. Codes Cryptogr., 63 (2012), 113-126.  doi: 10.1007/s10623-011-9539-4.

[19]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-Dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106. 

[20]

M. Greferath, Cyclic codes over finite rings, Discrete Mathematics, 177 (1997), 273-277.  doi: 10.1016/S0012-365X(97)00006-X.

[21]

T. A. Gulliver, Construction Of Quasi-Cyclic Codes, Ph. D. Dissertation, University of New Brunswick, 1984.

[22]

T. A. Gulliver and V. K. Bhargava, A (105, 10, 47) binary quasi-cyclic code, App. Math. Lett., 8 (1995), 67-70.  doi: 10.1016/0893-9659(95)00049-V.

[23]

S. Ling, H. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Des. Codes Cryptogr., 38 (2006), 337-361.  doi: 10.1007/s10623-005-1431-7.

[24]

S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅱ: Chain rings, Des. Codes Cryptogr., 30 (2003), 113-130.  doi: 10.1023/A:1024715527805.

[25]

E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Noncommutative rings and their applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 634 (2015), 227-241. doi: 10.1090/conm/634/12702.

[26]

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

[27]

J. Wolfmann, Negacyclic and cyclic codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 45 (1999), 2527-2532.  doi: 10.1109/18.796397.

[28]

J. Wood, Lecture Notes On Dual Codes And the MacWilliams Identities, Mexico, 2009.

[29]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, The American Journal of Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.

[30]

Magma computer algebra system, online, http://magma.maths.usyd.edu.au/.

[31]

, A Database on Binary Quasi-Cyclic Codes, online, Accessed January, 2018, http://www.tec.hkr.se/ chen/research/codes/qc.htm.

[32]

Code tables: Bounds on the parameters of codes, online, Accessed January, 2018, http://www.codetables.de/.

[33]

Database of $\mathbb{Z}_4$ codes, online, $Z_4$Codes.info, Accessed February, 2017.

Table 1.  Optimal binary linear codes
n g(x) or h(x) Binary Parameters
$ 8 $ $ g=x^2 + ux + 1 $ $ [32,24,4] $
$ 8 $ $ g=x^4+(uv+u)x^3+(v+u)x^2+ux+1 $ $ [32,16,8] $
$ 6 $ $ h=x^2+(u+v+1)x + 1 $ $ [24,8,8] $
$ 6 $ $ h=x+uv+1 $ $ [24,4,12] $
Table Options

Download as excel

n g(x) or h(x) Binary Parameters
$ 8 $ $ g=x^2 + ux + 1 $ $ [32,24,4] $
$ 8 $ $ g=x^4+(uv+u)x^3+(v+u)x^2+ux+1 $ $ [32,16,8] $
$ 6 $ $ h=x^2+(u+v+1)x + 1 $ $ [24,8,8] $
$ 6 $ $ h=x+uv+1 $ $ [24,4,12] $
Table 2.  Best-known binary linear codes
n g(x) or h(x) Binary Parameters
$ 12 $ $ h=x^5+(u+v+1)x^4+x^3+x^2+(uv+v+1)x+v+1 $ $ [48,20,12] $
$ 12 $ $ g=x^4+(uv+u+1)x^3+x+1 $ $ [48,32,6] $
$ 12 $ $ g=x^5+ux^4+x^3+ (u+v+ 1)x^2+v+1 $ $ [48,28,8] $
$ 16 $ $ h=x^4+vx^3+ux^2+1 $ $ [64,24,16] $
Table Options

Download as excel

n g(x) or h(x) Binary Parameters
$ 12 $ $ h=x^5+(u+v+1)x^4+x^3+x^2+(uv+v+1)x+v+1 $ $ [48,20,12] $
$ 12 $ $ g=x^4+(uv+u+1)x^3+x+1 $ $ [48,32,6] $
$ 12 $ $ g=x^5+ux^4+x^3+ (u+v+ 1)x^2+v+1 $ $ [48,28,8] $
$ 16 $ $ h=x^4+vx^3+ux^2+1 $ $ [64,24,16] $
Table 3.  New binary QC codes
n g(x) or h(x) Binary Parameters
$ 8 $ $ g=x^3+(uv+1)x^2+x+1 $ $ [32,20,4] $
$ 10 $ $ g=x^4+(u+v+ 1)x^3+x^2+x+1 $ $ [40,24,4] $
$ 12 $ $ g=x^3+ux^2+1 $ $ [48,36,4] $
$ 14 $ $ h=x^5+(uv+1)x^4+x^3+uvx^2+1 $ $ [56,20,7] $
$ 14 $ $ h=x^4+(u+v+1)x^3+x^2+1 $ $ [56,16,12] $
$ 14 $ $ g=x^3+ (u+v+1)x^2+1 $ $ [56,44,3] $
$ 14 $ $ h=x^3+ (u+v+1)x^2+1 $ $ [56,12,16] $
$ 16 $ $ h=x^6+(uv+v+1)x^4+ux^3+x^2+vx+uv+1 $ $ [64,24,16] $
$ 20 $ $ h=x^4+x^3+(u+1)x^2+x+u+v+1 $ $ [80,16,28] $
$ 18 $ $ h=x^4+(u+v+1)x^3+(u+v+1)x+1 $ $ [72,16,9] $
$ 28 $ $ h=x+u+v+1 $ $ [112,4,56] $
$ 28 $ $ h=x^4+x^2+(u+1)x+(u+1)v+1 $ $ [112,16,40] $
$ 30 $ $ h= x+uv+1 $ $ [120,4,60] $
$ 32 $ $ h=x+u+v+1 $ $ [128,4,64] $
$ 32 $ $ h=x^3+(u+1)x^2+x+1 $ $ [128,12,32] $
Table Options

Download as excel

n g(x) or h(x) Binary Parameters
$ 8 $ $ g=x^3+(uv+1)x^2+x+1 $ $ [32,20,4] $
$ 10 $ $ g=x^4+(u+v+ 1)x^3+x^2+x+1 $ $ [40,24,4] $
$ 12 $ $ g=x^3+ux^2+1 $ $ [48,36,4] $
$ 14 $ $ h=x^5+(uv+1)x^4+x^3+uvx^2+1 $ $ [56,20,7] $
$ 14 $ $ h=x^4+(u+v+1)x^3+x^2+1 $ $ [56,16,12] $
$ 14 $ $ g=x^3+ (u+v+1)x^2+1 $ $ [56,44,3] $
$ 14 $ $ h=x^3+ (u+v+1)x^2+1 $ $ [56,12,16] $
$ 16 $ $ h=x^6+(uv+v+1)x^4+ux^3+x^2+vx+uv+1 $ $ [64,24,16] $
$ 20 $ $ h=x^4+x^3+(u+1)x^2+x+u+v+1 $ $ [80,16,28] $
$ 18 $ $ h=x^4+(u+v+1)x^3+(u+v+1)x+1 $ $ [72,16,9] $
$ 28 $ $ h=x+u+v+1 $ $ [112,4,56] $
$ 28 $ $ h=x^4+x^2+(u+1)x+(u+1)v+1 $ $ [112,16,40] $
$ 30 $ $ h= x+uv+1 $ $ [120,4,60] $
$ 32 $ $ h=x+u+v+1 $ $ [128,4,64] $
$ 32 $ $ h=x^3+(u+1)x^2+x+1 $ $ [128,12,32] $
Table 4.  New quaternary codes
n g(x) or h(x) $ {\mathbb{Z}}_4 $ Parameters
$ 8 $ $ g=x^3+(u+1)x^2+3x+u+1 $ $ [24,10,9] $
$ 8 $ $ g=x^2+(3u+2)x+3u+3 $ $ [24,12,7] $
$ 8 $ $ g=x+u+1 $ $ [16,14,2] $
$ 12 $ $ g=x^2+(u+3)x+1 $ $ [36,20,8] $
$ 12 $ $ h=x^4+(3u+1)x^3+(u+2)x^2+(3u+3)x+3u +3 $ $ [24,8,12] $
$ 12 $ $ h=x^4+(3u+3)x^3+(u+2)x^2+(u+3)x+u+1 $ $ [36,8,20] $
$ 12 $ $ h=x^5+(2u+1)x^4+3x^3+(2u+1)x^2+3x+1 $ $ [24,10,8] $
$ 12 $ $ h=x^5+2ux^4+(3u+3)x^3+(2u+3)x^2+(3u+2)x+3 $ $ [36,10,17] $
$ 12 $ $ g=x^5+3ux^4+x^3+(3u+1)x^2+1 $ $ [36,14,13] $
$ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [28,8,18] $
$ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [42,8,26] $
$ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [28,6,18] $
$ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [42,6,29] $
$ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [28,20,6] $
$ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [42,20,13] $
$ 16 $ $ g=x^4 + (u + 2)x^3 + ux^2 + 2x + 1 $ $ [48,24,13] $
$ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [32,26,4] $
$ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [48,26,11] $
$ 16 $ $ g=x^2+(u+2)x+1 $ $ [32,28,2] $
$ 16 $ $ g=x^2+(u+2)x+1 $ $ [48,28,9] $
$ 18 $ $ g=x^4 + (2u + 3)x^3 + ux^2 + (2u + 1)x + 3u + 3 $ $ [54,28,13] $
$ 18 $ $ g=x^3 + ux^2 + 2x + 1 $ $ [54,30,10] $
$ 18 $ $ h=x^3+(3u+2)x^2+3ux +2u + 3 $ $ [54,6,34] $
$ 20 $ $ g=x^4 + 3x^3 + (3u + 1)x^2 + (3u + 1)x + 1 $ $ [60,32,13] $
$ 20 $ $ g=x^4 + (u + 3)x^3 + x^2 + x + 1 $ $ [40,32,4] $
$ 20 $ $ g=x^3 + (3u + 3)x^2 + (2u + 1)x + u + 3 $ $ [60,34,12] $
$ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [48,42,4] $
$ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [72,42,10] $
$ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [60,52,3] $
$ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [90,52,8] $
$ 32 $ $ g=x + u + 1 $ $ [64,62,2] $
$ 32 $ $ g=x + u + 1 $ $ [96,62,5] $
$ 32 $ $ h=x^4 + ux^2 + (u + 2)x + 3u + 1 $ $ [96,8,60] $
Table Options

Download as excel

n g(x) or h(x) $ {\mathbb{Z}}_4 $ Parameters
$ 8 $ $ g=x^3+(u+1)x^2+3x+u+1 $ $ [24,10,9] $
$ 8 $ $ g=x^2+(3u+2)x+3u+3 $ $ [24,12,7] $
$ 8 $ $ g=x+u+1 $ $ [16,14,2] $
$ 12 $ $ g=x^2+(u+3)x+1 $ $ [36,20,8] $
$ 12 $ $ h=x^4+(3u+1)x^3+(u+2)x^2+(3u+3)x+3u +3 $ $ [24,8,12] $
$ 12 $ $ h=x^4+(3u+3)x^3+(u+2)x^2+(u+3)x+u+1 $ $ [36,8,20] $
$ 12 $ $ h=x^5+(2u+1)x^4+3x^3+(2u+1)x^2+3x+1 $ $ [24,10,8] $
$ 12 $ $ h=x^5+2ux^4+(3u+3)x^3+(2u+3)x^2+(3u+2)x+3 $ $ [36,10,17] $
$ 12 $ $ g=x^5+3ux^4+x^3+(3u+1)x^2+1 $ $ [36,14,13] $
$ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [28,8,18] $
$ 14 $ $ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $ $ [42,8,26] $
$ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [28,6,18] $
$ 14 $ $ h=x^3+(u+3)x^2+(3u+2)x+u+1 $ $ [42,6,29] $
$ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [28,20,6] $
$ 14 $ $ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $ $ [42,20,13] $
$ 16 $ $ g=x^4 + (u + 2)x^3 + ux^2 + 2x + 1 $ $ [48,24,13] $
$ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [32,26,4] $
$ 16 $ $ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $ $ [48,26,11] $
$ 16 $ $ g=x^2+(u+2)x+1 $ $ [32,28,2] $
$ 16 $ $ g=x^2+(u+2)x+1 $ $ [48,28,9] $
$ 18 $ $ g=x^4 + (2u + 3)x^3 + ux^2 + (2u + 1)x + 3u + 3 $ $ [54,28,13] $
$ 18 $ $ g=x^3 + ux^2 + 2x + 1 $ $ [54,30,10] $
$ 18 $ $ h=x^3+(3u+2)x^2+3ux +2u + 3 $ $ [54,6,34] $
$ 20 $ $ g=x^4 + 3x^3 + (3u + 1)x^2 + (3u + 1)x + 1 $ $ [60,32,13] $
$ 20 $ $ g=x^4 + (u + 3)x^3 + x^2 + x + 1 $ $ [40,32,4] $
$ 20 $ $ g=x^3 + (3u + 3)x^2 + (2u + 1)x + u + 3 $ $ [60,34,12] $
$ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [48,42,4] $
$ 24 $ $ g=x^3 + 3ux^2 + (u + 2)x + 1 $ $ [72,42,10] $
$ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [60,52,3] $
$ 30 $ $ g=x^4 + (u + 1)x^3 + 1 $ $ [90,52,8] $
$ 32 $ $ g=x + u + 1 $ $ [64,62,2] $
$ 32 $ $ g=x + u + 1 $ $ [96,62,5] $
$ 32 $ $ h=x^4 + ux^2 + (u + 2)x + 3u + 1 $ $ [96,8,60] $
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