American Institute of Mathematical Sciences

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August  2018, 12(3): 595-605. doi: 10.3934/amc.2018035

A connection between sumsets and covering codes of a module

Otávio J. N. T. N. dos Santos 1, and  Emerson L. Monte Carmelo 2,,

1. 

Universidade Estadual de Mato Grosso do Sul, Unidade de Ponta Porã, Rua Itiberé Vieira, S/N, Ponta Porã-MS, CEP 79907-414, Brazil
2. 

Universidade Estadual de Maringá, Departamento de Matemática, Av. Colombo, 5790 - Campus Universitário, Maringá-PR, CEP 87020-900, Brazil

* Corresponding author: elmcarmelo@uem.br

Received  October 2017 Published  July 2018

Fund Project: The first author is partially supported by FUNDECT/CNPq grant 054/2015
The second author is partially supported by CNPq/MCT grant 311793/2016-0.

In this work we focus on a connection between sumsets and covering codes in an arbitrary finite module. For this purpose, bounds on a new problem on sumsets are obtained from well-known results of additive number theory, namely, the Cauchy-Davenport theorem, the Vosper theorem and a theorem due to Hamidoune-Rødseth. As an application, the approach is able to extend the Blokhuis-Lam theorems and a construction of covering codes by Honkala to an arbitrary module.

Keywords: Sumset, arithmetic progression, covering code, code over a ring, matrix method, bound on code.
Mathematics Subject Classification: Primary: 11T71; Secondary: 11P70, 13M99, 94B75.
Citation: Otávio J. N. T. N. dos Santos, Emerson L. Monte Carmelo. A connection between sumsets and covering codes of a module. Advances in Mathematics of Communications, 2018, 12 (3) : 595-605. doi: 10.3934/amc.2018035
References:
[1]

A. Blokhuis and C. W. H. Lam, More covering by rook domains, J. Combin. Theory Ser. A, 36 (1984), 240-244.  doi: 10.1016/0097-3165(84)90010-4.  Google Scholar

[2]

W. A. Carnielli, On covering and coloring problems for rook domains, Discrete Math., 57 (1985), 9-16.  doi: 10.1016/0012-365X(85)90152-9.  Google Scholar

[3]

W. A. Carnielli, Hyper-rook domain inequalities, Stud. Appl. Math., 82 (1990), 59-69.  doi: 10.1002/sapm199082159.  Google Scholar

[4]

A. Cauchy, Recherches sur les nombres, Oeuvres completes, (2009), 39-63.  doi: 10.1017/CBO9780511702501.004.  Google Scholar

[5]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997.  Google Scholar

[6]

H. Davenport, On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.   Google Scholar

[7]

O. J. N. T. N. dos Santos and E. L. Monte Carmelo, Invariant sets under linear operator and covering codes over modules, Linear Algebra Appl., 444 (2014), 42-52.  doi: 10.1016/j.laa.2013.11.034.  Google Scholar

[8]

P. Erdős, Problems and Results on Combinatorial Number Theory Ⅲ, in Lecture Notes in Math., Springer- Verlag, Berlin, (1977), 43-72.  Google Scholar

[9]

Y. O. Hamidoune and Ø. J. Rødseth, An inverse theorem mod $p$, Acta Arithmetica, 92 (2000), 251-262.  doi: 10.4064/aa-92-3-251-262.  Google Scholar

[10]

I. Honkala, On lengthening of covering codes, Discrete Math., 106/107 (1992), 291-295.  doi: 10.1016/0012-365X(92)90556-U.  Google Scholar

[11]

H. J. L. Kamps and J. H. van Lint, A covering problem, in Combinatorial theory and its applications Ⅱ (Colloquia Mathematica Societatis Jànos Bolyai), North-Holland, Amsterdam, (1970), 679-685.  Google Scholar

[12]

I. N. Nakaoka and O. J. N. T. N. dos Santos, A covering problem over finite rings, Appl. Math. Lett., 23 (2010), 322-326.  doi: 10.1016/j.aml.2009.09.022.  Google Scholar

[13]

I. N. Nakaoka, E. L. Monte Carmelo and O. J. N. T. N. dos Santos, Sharp covering of a module by cyclic submodules, Linear Algebra Appl., 458 (2014), 387-402.  doi: 10.1016/j.laa.2014.06.019.  Google Scholar

[14]

M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[15]

T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105 Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149.  Google Scholar

[16]

O. Taussky and J. Todd, Covering theorems for groups, Ann. Soc. Polonaise Math., 21 (1948), 303-305.   Google Scholar

[17]

J. H. van Lint Jr., Covering Radius Problems, M. Sc. Thesis, Eindhoven University of Technology, The Netherlands, 1988. Google Scholar

[18]

G. Vosper, The critical pairs of subsets of a group of prime order, J. London Math. Soc., 31 (1956), 200-205.  doi: 10.1112/jlms/s1-31.2.200.  Google Scholar

show all references

References:
[1]

A. Blokhuis and C. W. H. Lam, More covering by rook domains, J. Combin. Theory Ser. A, 36 (1984), 240-244.  doi: 10.1016/0097-3165(84)90010-4.  Google Scholar

[2]

W. A. Carnielli, On covering and coloring problems for rook domains, Discrete Math., 57 (1985), 9-16.  doi: 10.1016/0012-365X(85)90152-9.  Google Scholar

[3]

W. A. Carnielli, Hyper-rook domain inequalities, Stud. Appl. Math., 82 (1990), 59-69.  doi: 10.1002/sapm199082159.  Google Scholar

[4]

A. Cauchy, Recherches sur les nombres, Oeuvres completes, (2009), 39-63.  doi: 10.1017/CBO9780511702501.004.  Google Scholar

[5]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997.  Google Scholar

[6]

H. Davenport, On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.   Google Scholar

[7]

O. J. N. T. N. dos Santos and E. L. Monte Carmelo, Invariant sets under linear operator and covering codes over modules, Linear Algebra Appl., 444 (2014), 42-52.  doi: 10.1016/j.laa.2013.11.034.  Google Scholar

[8]

P. Erdős, Problems and Results on Combinatorial Number Theory Ⅲ, in Lecture Notes in Math., Springer- Verlag, Berlin, (1977), 43-72.  Google Scholar

[9]

Y. O. Hamidoune and Ø. J. Rødseth, An inverse theorem mod $p$, Acta Arithmetica, 92 (2000), 251-262.  doi: 10.4064/aa-92-3-251-262.  Google Scholar

[10]

I. Honkala, On lengthening of covering codes, Discrete Math., 106/107 (1992), 291-295.  doi: 10.1016/0012-365X(92)90556-U.  Google Scholar

[11]

H. J. L. Kamps and J. H. van Lint, A covering problem, in Combinatorial theory and its applications Ⅱ (Colloquia Mathematica Societatis Jànos Bolyai), North-Holland, Amsterdam, (1970), 679-685.  Google Scholar

[12]

I. N. Nakaoka and O. J. N. T. N. dos Santos, A covering problem over finite rings, Appl. Math. Lett., 23 (2010), 322-326.  doi: 10.1016/j.aml.2009.09.022.  Google Scholar

[13]

I. N. Nakaoka, E. L. Monte Carmelo and O. J. N. T. N. dos Santos, Sharp covering of a module by cyclic submodules, Linear Algebra Appl., 458 (2014), 387-402.  doi: 10.1016/j.laa.2014.06.019.  Google Scholar

[14]

M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[15]

T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105 Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149.  Google Scholar

[16]

O. Taussky and J. Todd, Covering theorems for groups, Ann. Soc. Polonaise Math., 21 (1948), 303-305.   Google Scholar

[17]

J. H. van Lint Jr., Covering Radius Problems, M. Sc. Thesis, Eindhoven University of Technology, The Netherlands, 1988. Google Scholar

[18]

G. Vosper, The critical pairs of subsets of a group of prime order, J. London Math. Soc., 31 (1956), 200-205.  doi: 10.1112/jlms/s1-31.2.200.  Google Scholar

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