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May  2017, 11(2): 329-338. doi: 10.3934/amc.2017025

On the covering radius of some binary cyclic codes

Rafael Arce-Nazario 1, , Francis N. Castro 2,, and  Jose Ortiz-Ubarri 1,

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

* Corresponding author

Received  February 2016 Revised  March 2016 Published  May 2017

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 $f=7, 8$.

Keywords: Binary cyclic codes, covering radius, minimum distance.
Mathematics Subject Classification: Primary: 94B15; Secondary: 11T71.
Citation: Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
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. 

Figure 1.  Construction of a solution for $\alpha_1 = 1 \in \mathbb{F}_{2^7}$. By choosing numbers that have an even quantity of 1's for all columns except the least significant, we guarantee that solution $\alpha_1 = 0000001$. The values for $\left(x_1,\ldots,x_5\right)$ are read from the rows
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Table 1.  Codes ${\mathcal C}_{1,d}$ with minimum distance $\geq 4$ and covering radius 3
${\mathcal C}_{1,d}/\mathbb{F}_{128}$ ${\mathcal C}_{1,d}/\mathbb{F}_{512}$ ${\mathcal C}_{1,d}/\mathbb{F}_{2048}$
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
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${\mathcal C}_{1,d}/\mathbb{F}_{128}$ ${\mathcal C}_{1,d}/\mathbb{F}_{512}$ ${\mathcal C}_{1,d}/\mathbb{F}_{2048}$
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
Table 2.  Codes ${\mathcal C}_{1,d}$ with minimum distance $\geq 4$ and covering radius 3
${\mathcal C}_{1,d}/\mathbb{F}_{2^{13}}$
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
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${\mathcal C}_{1,d}/\mathbb{F}_{2^{13}}$
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
Table 3.  Covering radius of binary primitive $BCH$ codes
$n$ $BCH(e)$Covering Radius
127 $BCH(3)$5
127 $BCH(4)$7
127 $BCH(5)$9
255 $BCH(3)$5
255 $BCH(4)$7
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$n$ $BCH(e)$Covering Radius
127 $BCH(3)$5
127 $BCH(4)$7
127 $BCH(5)$9
255 $BCH(3)$5
255 $BCH(4)$7
Table 4.  Covering radius of cyclic codes of type ${\mathcal C}_{1, 2^i+1, 2^{2i}+1}$
$n$ ${\mathcal C}_{1, 2^i+1, 2^{2i}+1}$Covering Radius
31 ${\mathcal C}_{1, 5, 17}$5
127 ${\mathcal C}_{1, 5, 17}$5
127 ${\mathcal C}_{1, 9, 65}$5
255 ${\mathcal C}_{1, 9, 65}$5
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$n$ ${\mathcal C}_{1, 2^i+1, 2^{2i}+1}$Covering Radius
31 ${\mathcal C}_{1, 5, 17}$5
127 ${\mathcal C}_{1, 5, 17}$5
127 ${\mathcal C}_{1, 9, 65}$5
255 ${\mathcal C}_{1, 9, 65}$5
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