# 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
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##### References:
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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