# American Institute of Mathematical Sciences

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

## 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

* 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 $f=7, 8$.

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.pdfGoogle Scholar [2] C. Bracken and T. Helleseth, Triple-error-correcting BCH-like codes, in Proc. 2009 IEEE Int. Conf. Symp. Inf. Theory, 2009,1723-1725.Google Scholar [3] R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998,755-826. Google Scholar [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. Google Scholar [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.Google Scholar [6] P. Charpin, A. Tietäväinen and V. Zinoviev, On binary cyclic codes with minimum distance d=3, Probl. Inform. Transm., 33 (1997), 287-296. Google Scholar [7] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland Math. Library, Elsevier. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. Google Scholar [12] T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in Proc. Conf. Combinat. Math. Appl. , Univ. North Carolina, Chapel Hill, 1969,335-357. Google Scholar [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. Google Scholar [14] O. Moreno, Further results on quasi-perfect codes related to the Goppa codes, Congresus Numerant., 40 (1983), 249-256. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] T. J. Wagner, A search technique for quasi-perfect, Inf. Control, 9 (1966), 94-99. Google Scholar

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##### 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.pdfGoogle Scholar [2] C. Bracken and T. Helleseth, Triple-error-correcting BCH-like codes, in Proc. 2009 IEEE Int. Conf. Symp. Inf. Theory, 2009,1723-1725.Google Scholar [3] R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998,755-826. Google Scholar [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. Google Scholar [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.Google Scholar [6] P. Charpin, A. Tietäväinen and V. Zinoviev, On binary cyclic codes with minimum distance d=3, Probl. Inform. Transm., 33 (1997), 287-296. Google Scholar [7] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland Math. Library, Elsevier. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. Google Scholar [12] T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in Proc. Conf. Combinat. Math. Appl. , Univ. North Carolina, Chapel Hill, 1969,335-357. Google Scholar [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. Google Scholar [14] O. Moreno, Further results on quasi-perfect codes related to the Goppa codes, Congresus Numerant., 40 (1983), 249-256. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] T. J. Wagner, A search technique for quasi-perfect, Inf. Control, 9 (1966), 94-99. Google Scholar
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
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, 1521, 23, 27, 29, 43 3, 5, 13, 17, 19, 2731, 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
 ${\mathcal C}_{1,d}/\mathbb{F}_{128}$ ${\mathcal C}_{1,d}/\mathbb{F}_{512}$ ${\mathcal C}_{1,d}/\mathbb{F}_{2048}$ 3, 5, 9, 11, 13, 1521, 23, 27, 29, 43 3, 5, 13, 17, 19, 2731, 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
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
 ${\mathcal C}_{1,d}/\mathbb{F}_{2^{13}}$ 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
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
 $n$ $BCH(e)$ Covering Radius 127 $BCH(3)$ 5 127 $BCH(4)$ 7 127 $BCH(5)$ 9 255 $BCH(3)$ 5 255 $BCH(4)$ 7
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
 $n$ ${\mathcal C}_{1, 2^i+1, 2^{2i}+1}$ 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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