Codes over local rings of order 16 and binary codes

Previous Article
Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code
AMC Home
This Issue
Next Article
The geometric structure of relative one-weight codes

May 2016, 10(2): 379-391. doi: 10.3934/amc.2016012

Codes over local rings of order 16 and binary codes

Steven T. Dougherty ^1, , Esengül Saltürk ^1, and Steve Szabo ^2,

1.	Department of Mathematics, University of Scranton, Scranton, PA 18510, United States, United States
2.	Department of Mathematics and Statistics, Eastern Kentucky University Richmond, KY 40475, United States

Received August 2014 Revised February 2015 Published April 2016

Abstract
Related Papers

We study codes over the commutative local Frobenius rings of order 16 with maximal ideals of size 8. We define a weight preserving Gray map and study the images of these codes as binary codes. We study self-dual codes and determine when they exist.

Keywords: Gray maps., Codes over rings.

Mathematics Subject Classification: Primary: 11T71, 94B05; Secondary: 13H9.

Citation: Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012

References:

[1]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE-IT, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[2]	S. T. Dougherty and C. Fernandez-Cordoba, Codes over $\mathbbZ_{2^k}$, Gray maps and self-dual codes, Adv. Math. Commun., 5 (2011), 571-588. doi: 10.3934/amc.2011.5.571.
[3]	S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa and P. Solé, Type IV self-dual codes over rings, IEEE-IT, 45 (1999), 2345-2360. doi: 10.1109/18.796375.
[4]	S. T. Dougherty, J. L. Kim, H. Kulosman and H. Liu, Self-dual codes over Frobenius rings, Finite Fields Appl., 16 (2010), 14-26. doi: 10.1016/j.ffa.2009.11.004.
[5]	S. T. Dougherty and H. Liu, Independence of vectors in codes over rings, Des. Codes Crypt., 51 (2009), 55-68. doi: 10.1007/s10623-008-9243-1.
[6]	S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE-IT, 47 (2001), 400-404. doi: 10.1109/18.904544.
[7]	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.
[8]	A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE-IT, 40 (1994), 301-319. doi: 10.1109/18.312154.
[9]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[10]	E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Contemp. Math., 634 (2015), 227-242 doi: 10.1090/conm/634/12702.
[11]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.
[12]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbbF_2 + u \mathbbF_2 + v \mathbbF_2 + uv \mathbbF_2$, Des. Codes Crypt., 54 (2010), 61-81. doi: 10.1007/s10623-009-9309-8.
[13]	B. Yildiz and S. Karadeniz, A new construction for the extended binary Golay code, Appl. Math. Inf. Sci., 8 (2014), 69-72. doi: 10.12785/amis/080107.
[14]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbbZ_4 + u \mathbbZ_4$, MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27 (2014), 24-40. doi: 10.1016/j.ffa.2013.12.007.

show all references

References:

[1]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE-IT, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[2]	S. T. Dougherty and C. Fernandez-Cordoba, Codes over $\mathbbZ_{2^k}$, Gray maps and self-dual codes, Adv. Math. Commun., 5 (2011), 571-588. doi: 10.3934/amc.2011.5.571.
[3]	S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa and P. Solé, Type IV self-dual codes over rings, IEEE-IT, 45 (1999), 2345-2360. doi: 10.1109/18.796375.
[4]	S. T. Dougherty, J. L. Kim, H. Kulosman and H. Liu, Self-dual codes over Frobenius rings, Finite Fields Appl., 16 (2010), 14-26. doi: 10.1016/j.ffa.2009.11.004.
[5]	S. T. Dougherty and H. Liu, Independence of vectors in codes over rings, Des. Codes Crypt., 51 (2009), 55-68. doi: 10.1007/s10623-008-9243-1.
[6]	S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE-IT, 47 (2001), 400-404. doi: 10.1109/18.904544.
[7]	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.
[8]	A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE-IT, 40 (1994), 301-319. doi: 10.1109/18.312154.
[9]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[10]	E. Martinez-Moro and S. Szabo, On codes over local Frobenius non-chain rings of order 16, Contemp. Math., 634 (2015), 227-242 doi: 10.1090/conm/634/12702.
[11]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.
[12]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbbF_2 + u \mathbbF_2 + v \mathbbF_2 + uv \mathbbF_2$, Des. Codes Crypt., 54 (2010), 61-81. doi: 10.1007/s10623-009-9309-8.
[13]	B. Yildiz and S. Karadeniz, A new construction for the extended binary Golay code, Appl. Math. Inf. Sci., 8 (2014), 69-72. doi: 10.12785/amis/080107.
[14]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbbZ_4 + u \mathbbZ_4$, MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27 (2014), 24-40. doi: 10.1016/j.ffa.2013.12.007.

[1]	Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[2]	Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022002
[3]	Hai Quang Dinh, Atul Gaur, Pratyush Kumar, Manoj Kumar Singh, Abhay Kumar Singh. Cyclic codes over rings of matrices. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022073
[4]	Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[5]	Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[6]	Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[7]	Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[8]	Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[9]	Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022028
[10]	Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[11]	Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[12]	Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[13]	Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045
[14]	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
[15]	Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[16]	Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[17]	David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[18]	Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[19]	Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99
[20]	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

2021 Impact Factor: 1.015

Tools

Metrics

Other articles
by authors

on AIMS
Steven T. Dougherty Esengül Saltürk Steve Szabo
on Google Scholar
Steven T. Dougherty Esengül Saltürk Steve Szabo

[Back to Top]