
-
Previous Article
On constructions of bent, semi-bent and five valued spectrum functions from old bent functions
- AMC Home
- This Issue
-
Next Article
Arrays composed from the extended rational cycle
On the covering radius of some binary cyclic codes
1. | Department of Computer Science, University of Puerto Rico, Río Piedras, San Juan, PR 00931 |
2. | Department of Mathematics, University of Puerto Rico, Río Piedras, San Juan, PR 00931 |
We compute the covering radius of some families of binary cyclic codes. In particular, we compute the covering radius of cyclic codes with two zeros and minimum distance greater than 3. We also compute the covering radius of some binary primitive BCH codes over $\mathbb{F}_{2^f}$, where $f=7, 8$.
References:
[1] |
R. Arce-Nazario, F. N. Castro and J. Ortiz-Ubarri,
On the covering radius of some binary cyclic codes available at https://franciscastr.files.wordpress.com/2016/01/coverig-radius2016.pdf |
[2] |
C. Bracken and T. Helleseth, Triple-error-correcting BCH-like codes, in Proc. 2009 IEEE Int.
Conf. Symp. Inf. Theory, 2009,1723-1725. |
[3] |
R. A. Brualdi, S. Litsyn and V. S. Pless,
Covering radius, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998,755-826. |
[4] |
C. Carlet, P. Charpin and V. Zinoviev,
Bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-155.
doi: 10.1023/A:1008344232130. |
[5] |
F. N. Castro, I. Rubio, H. Randriam, O. Moreno and H. F. Mattson, Jr. , An elementary
approach to Ax-Katz, McEliece's divisibility and applications to quasi-perfect binary 2-error
correcting codes, in Proc. 2006 IEEE Int. Symp. Inf. Theory, Seattle, 2006,1905-1908. |
[6] |
P. Charpin, A. Tietäväinen and V. Zinoviev,
On binary cyclic codes with minimum distance d=3, Probl. Inform. Transm., 33 (1997), 287-296.
|
[7] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein,
Covering Codes, North-Holland Math. Library, Elsevier. |
[8] |
T. Etzion and G. Greenberg,
Constructions for perfect mixed codes and other codes, IEEE Trans. Inf. Theory, 39 (1993), 209-214.
doi: 10.1109/18.179360. |
[9] |
T. Etzion and B. Mounits,
Quasi-perfect codes with small distance, IEEE Trans. Inf. Theory, 51 (2005), 3938-3946.
doi: 10.1109/TIT.2005.856944. |
[10] |
E. M. Gorenstein, W. W. Peterson and N. Zierler,
Two-error correcting Bose-Chaudhuri codes are quasi-perfect, Inf. Contr., 3 (1960), 291-294.
|
[11] |
W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[12] |
T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in Proc. Conf. Combinat. Math. Appl. , Univ. North Carolina, Chapel Hill, 1969,335-357. |
[13] |
T. Kasami,
The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Inf. Control, 18 (1971), 369-394.
|
[14] |
O. Moreno,
Further results on quasi-perfect codes related to the Goppa codes, Congresus Numerant., 40 (1983), 249-256.
|
[15] |
O. Moreno and F. Castro,
Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inf. Theory, 49 (2003), 3299-3303.
doi: 10.1109/TIT.2003.820033. |
[16] |
O. Moreno and F. Castro, On the covering radius of certain cyclic codes, in Applied Algebra, Algebraic Algorithms and Error Correcting Codes, 2003,129-138.
doi: 10.1007/3-540-44828-4_15. |
[17] |
O. Moreno and F. Castro,
Improvement of Ax-Katz's and Moreno-Moreno's results and applications, Int. J. Pure Appl. Math., 19 (2005), 259-267.
|
[18] |
O. Moreno, F. Castro and H. F. Mattson Jr.,
Correction, divisibility properties for covering radius for certain cyclic codes, IEEE Trans. Inf. Theory, 52 (2006), 1798-1799.
doi: 10.1109/TIT.2003.820033. |
[19] |
O. Moreno and C. J. Moreno,
Improvement of the Chevalley-Warning and the Ax-Katz theorems, Amer. J. Math., 117 (1995), 241-244.
doi: 10.2307/2375042. |
[20] |
O. Moreno, K. Shum, F. N. Castro and P. V. Kumar,
Tight bounds for Chevalley-Warning-Ax type estimates, with improved applications, Proc. London Math. Soc., 88 (2004), 545-564.
doi: 10.1112/S002461150301462X. |
[21] |
T. J. Wagner,
A search technique for quasi-perfect, Inf. Control, 9 (1966), 94-99.
|
show all references
References:
[1] |
R. Arce-Nazario, F. N. Castro and J. Ortiz-Ubarri,
On the covering radius of some binary cyclic codes available at https://franciscastr.files.wordpress.com/2016/01/coverig-radius2016.pdf |
[2] |
C. Bracken and T. Helleseth, Triple-error-correcting BCH-like codes, in Proc. 2009 IEEE Int.
Conf. Symp. Inf. Theory, 2009,1723-1725. |
[3] |
R. A. Brualdi, S. Litsyn and V. S. Pless,
Covering radius, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998,755-826. |
[4] |
C. Carlet, P. Charpin and V. Zinoviev,
Bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-155.
doi: 10.1023/A:1008344232130. |
[5] |
F. N. Castro, I. Rubio, H. Randriam, O. Moreno and H. F. Mattson, Jr. , An elementary
approach to Ax-Katz, McEliece's divisibility and applications to quasi-perfect binary 2-error
correcting codes, in Proc. 2006 IEEE Int. Symp. Inf. Theory, Seattle, 2006,1905-1908. |
[6] |
P. Charpin, A. Tietäväinen and V. Zinoviev,
On binary cyclic codes with minimum distance d=3, Probl. Inform. Transm., 33 (1997), 287-296.
|
[7] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein,
Covering Codes, North-Holland Math. Library, Elsevier. |
[8] |
T. Etzion and G. Greenberg,
Constructions for perfect mixed codes and other codes, IEEE Trans. Inf. Theory, 39 (1993), 209-214.
doi: 10.1109/18.179360. |
[9] |
T. Etzion and B. Mounits,
Quasi-perfect codes with small distance, IEEE Trans. Inf. Theory, 51 (2005), 3938-3946.
doi: 10.1109/TIT.2005.856944. |
[10] |
E. M. Gorenstein, W. W. Peterson and N. Zierler,
Two-error correcting Bose-Chaudhuri codes are quasi-perfect, Inf. Contr., 3 (1960), 291-294.
|
[11] |
W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[12] |
T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in Proc. Conf. Combinat. Math. Appl. , Univ. North Carolina, Chapel Hill, 1969,335-357. |
[13] |
T. Kasami,
The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Inf. Control, 18 (1971), 369-394.
|
[14] |
O. Moreno,
Further results on quasi-perfect codes related to the Goppa codes, Congresus Numerant., 40 (1983), 249-256.
|
[15] |
O. Moreno and F. Castro,
Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inf. Theory, 49 (2003), 3299-3303.
doi: 10.1109/TIT.2003.820033. |
[16] |
O. Moreno and F. Castro, On the covering radius of certain cyclic codes, in Applied Algebra, Algebraic Algorithms and Error Correcting Codes, 2003,129-138.
doi: 10.1007/3-540-44828-4_15. |
[17] |
O. Moreno and F. Castro,
Improvement of Ax-Katz's and Moreno-Moreno's results and applications, Int. J. Pure Appl. Math., 19 (2005), 259-267.
|
[18] |
O. Moreno, F. Castro and H. F. Mattson Jr.,
Correction, divisibility properties for covering radius for certain cyclic codes, IEEE Trans. Inf. Theory, 52 (2006), 1798-1799.
doi: 10.1109/TIT.2003.820033. |
[19] |
O. Moreno and C. J. Moreno,
Improvement of the Chevalley-Warning and the Ax-Katz theorems, Amer. J. Math., 117 (1995), 241-244.
doi: 10.2307/2375042. |
[20] |
O. Moreno, K. Shum, F. N. Castro and P. V. Kumar,
Tight bounds for Chevalley-Warning-Ax type estimates, with improved applications, Proc. London Math. Soc., 88 (2004), 545-564.
doi: 10.1112/S002461150301462X. |
[21] |
T. J. Wagner,
A search technique for quasi-perfect, Inf. Control, 9 (1966), 94-99.
|

| | |
3, 5, 9, 11, 13, 15 21, 23, 27, 29, 43 |
3, 5, 13, 17, 19, 27 31, 43, 47, 87 |
3, 5, 9, 11, 13, 17, 25, 33, 35, 37, 43, 47, 49 57, 63, 81, 87, 95,105,121,139,141,143,151 171,187,189,206,221,229,231,249 295,311,315,343,363,365,413,429 |
| | |
3, 5, 9, 11, 13, 15 21, 23, 27, 29, 43 |
3, 5, 13, 17, 19, 27 31, 43, 47, 87 |
3, 5, 9, 11, 13, 17, 25, 33, 35, 37, 43, 47, 49 57, 63, 81, 87, 95,105,121,139,141,143,151 171,187,189,206,221,229,231,249 295,311,315,343,363,365,413,429 |
3, 5, 9, 11, 13, 17, 19, 21, 33, 43, 57, 65, 67, 71, 95, 97,113,127,129,147,161,171,191,205,225,241,287,347,363,367,405,483,485,497,631,635,683,745,747,749,869,911,919,939,949,953,973,1367,1453,1461,1639,1643,1645,1691,1707,2047,2731 |
3, 5, 9, 11, 13, 17, 19, 21, 33, 43, 57, 65, 67, 71, 95, 97,113,127,129,147,161,171,191,205,225,241,287,347,363,367,405,483,485,497,631,635,683,745,747,749,869,911,919,939,949,953,973,1367,1453,1461,1639,1643,1645,1691,1707,2047,2731 |
| Covering Radius | |
127 | | 5 |
127 | | 7 |
127 | | 9 |
255 | | 5 |
255 | | 7 |
| Covering Radius | |
127 | | 5 |
127 | | 7 |
127 | | 9 |
255 | | 5 |
255 | | 7 |
| Covering Radius | |
31 | | 5 |
127 | | 5 |
127 | | 5 |
255 | | 5 |
| Covering Radius | |
31 | | 5 |
127 | | 5 |
127 | | 5 |
255 | | 5 |
[1] |
San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038 |
[2] |
Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399 |
[3] |
José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018 |
[4] |
Manish K. Gupta, Chinnappillai Durairajan. On the covering radius of some modular codes. Advances in Mathematics of Communications, 2014, 8 (2) : 129-137. doi: 10.3934/amc.2014.8.129 |
[5] |
Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021043 |
[6] |
Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175 |
[7] |
Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65 |
[8] |
Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2021074 |
[9] |
Rong Wang, Xiaoni Du, Cuiling Fan. Infinite families of 2-designs from a class of non-binary Kasami cyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 663-676. doi: 10.3934/amc.2020088 |
[10] |
Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang. Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros. Advances in Mathematics of Communications, 2022, 16 (1) : 157-168. doi: 10.3934/amc.2020106 |
[11] |
Akbar Mahmoodi Rishakani, Seyed Mojtaba Dehnavi, Mohmmadreza Mirzaee Shamsabad, Nasour Bagheri. Cryptographic properties of cyclic binary matrices. Advances in Mathematics of Communications, 2021, 15 (2) : 311-327. doi: 10.3934/amc.2020068 |
[12] |
Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83 |
[13] |
Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225 |
[14] |
Otávio J. N. T. N. dos Santos, Emerson L. Monte Carmelo. A connection between sumsets and covering codes of a module. Advances in Mathematics of Communications, 2018, 12 (3) : 595-605. doi: 10.3934/amc.2018035 |
[15] |
Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195 |
[16] |
Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 |
[17] |
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 |
[18] |
Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177 |
[19] |
Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329 |
[20] |
Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]