Codes over local rings of order 16 and binary codes

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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
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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.

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