On the covering radius of some binary cyclic codes

Rafael Arce-Nazario; Francis N. Castro; Jose Ortiz-Ubarri

doi:10.3934/amc.2017025

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2017, Volume 11, Issue 2: 329-338. Doi: 10.3934/amc.2017025

This issue Previous Article Arrays composed from the extended rational cycle Next Article On constructions of bent, semi-bent and five valued spectrum functions from old bent functions

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
* Corresponding author

Received: January 31, 2016

Revised: February 29, 2016

Published: September 2017

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Abstract

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 $f=7, 8$.

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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

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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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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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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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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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