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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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

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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