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Some generalizations of good integers and their applications in the study of self-dual negacyclic codes
Skew constacyclic codes over the local Frobenius non-chain rings of order 16
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 |
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.
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. |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | Binary Parameters |
n | g(x) or h(x) | |
n | g(x) or h(x) | |
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