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A connection between sumsets and covering codes of a module
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 |
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.
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. |
[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. |
[3] |
W. A. Carnielli,
Hyper-rook domain inequalities, Stud. Appl. Math., 82 (1990), 59-69.
doi: 10.1002/sapm199082159. |
[4] |
A. Cauchy,
Recherches sur les nombres, Oeuvres completes, (2009), 39-63.
doi: 10.1017/CBO9780511702501.004. |
[5] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[6] |
H. Davenport,
On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.
|
[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. |
[8] |
P. Erdős, Problems and Results on Combinatorial Number Theory Ⅲ, in Lecture Notes in Math., Springer- Verlag, Berlin, (1977), 43-72. |
[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. |
[10] |
I. Honkala,
On lengthening of covering codes, Discrete Math., 106/107 (1992), 291-295.
doi: 10.1016/0012-365X(92)90556-U. |
[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. |
[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. |
[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. |
[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. |
[15] |
T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105 Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511755149. |
[16] |
O. Taussky and J. Todd,
Covering theorems for groups, Ann. Soc. Polonaise Math., 21 (1948), 303-305.
|
[17] |
J. H. van Lint Jr., Covering Radius Problems, M. Sc. Thesis, Eindhoven University of Technology, The Netherlands, 1988. |
[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. |
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. |
[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. |
[3] |
W. A. Carnielli,
Hyper-rook domain inequalities, Stud. Appl. Math., 82 (1990), 59-69.
doi: 10.1002/sapm199082159. |
[4] |
A. Cauchy,
Recherches sur les nombres, Oeuvres completes, (2009), 39-63.
doi: 10.1017/CBO9780511702501.004. |
[5] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[6] |
H. Davenport,
On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.
|
[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. |
[8] |
P. Erdős, Problems and Results on Combinatorial Number Theory Ⅲ, in Lecture Notes in Math., Springer- Verlag, Berlin, (1977), 43-72. |
[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. |
[10] |
I. Honkala,
On lengthening of covering codes, Discrete Math., 106/107 (1992), 291-295.
doi: 10.1016/0012-365X(92)90556-U. |
[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. |
[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. |
[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. |
[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. |
[15] |
T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105 Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511755149. |
[16] |
O. Taussky and J. Todd,
Covering theorems for groups, Ann. Soc. Polonaise Math., 21 (1948), 303-305.
|
[17] |
J. H. van Lint Jr., Covering Radius Problems, M. Sc. Thesis, Eindhoven University of Technology, The Netherlands, 1988. |
[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. |
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