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# On the covering radius of some binary cyclic codes

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• 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 $f=7, 8$.

Mathematics Subject Classification: Primary: 94B15; Secondary: 11T71.

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• 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

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, 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

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

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

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
•  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 C. Bracken and T. Helleseth, Triple-error-correcting BCH-like codes, in Proc. 2009 IEEE Int. Conf. Symp. Inf. Theory, 2009,1723-1725. R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998,755-826. 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. 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. P. Charpin , A. Tietäväinen  and  V. Zinoviev , On binary cyclic codes with minimum distance d=3, Probl. Inform. Transm., 33 (1997) , 287-296. G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland Math. Library, Elsevier. 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. 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. E. M. Gorenstein , W. W. Peterson  and  N. Zierler , Two-error correcting Bose-Chaudhuri codes are quasi-perfect, Inf. Contr., 3 (1960) , 291-294. W. Huffman and  V. Pless,  Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077. T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem codes, in Proc. Conf. Combinat. Math. Appl. , Univ. North Carolina, Chapel Hill, 1969,335-357. T. Kasami , The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Inf. Control, 18 (1971) , 369-394. O. Moreno , Further results on quasi-perfect codes related to the Goppa codes, Congresus Numerant., 40 (1983) , 249-256. 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. 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. 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. 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. 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. 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. T. J. Wagner , A search technique for quasi-perfect, Inf. Control, 9 (1966) , 94-99.

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