g-Golomb Rulers

YADIRA CAICEDO^{a *}, CARLOS A. MARTOS^b , CARLOS A. TRUJILLO^b

^aUniversidad del Tolima, Departamento de Matemáticas y Estadística, Ibagué, Colombia.

^bUniversidad del Cauca, Departamento de Matemáticas, Popayán, Colombia.

Abstract. A set of positive integers A is called a g-Golomb ruler if the difference between two distinct elements of A is repeated at most g times. This definition is a generalization of the Golomb ruler (g = 1). In this paper we construct g-Golomb ruler from Golomb ruler and we prove two theorems about extremal functions associated with this sets.

Keywords: Sidon sets, B₂ sets, Golomb ruler.
MSC2010: 11B50, 12E20, 20K01, 20K30.

Reglas g-Golomb

Resumen. Se dice que un conjunto de enteros positivos A satisface la regla g-Golomb si la diferencia entre dos elementos distintos de A se repite a lo más g veces. Esta definición es una generalización de las reglas de Golomb (g = 1). En este artículo construimos reglas g-Golomb a partir de reglas Golomb y demostramos dos teoremas sobre las funciones extremas asociadas con estos conjuntos.

Palabras clave: Conjuntos de Sidon, conjuntos B₂, reglas Golomb.

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Referencias

[1] Atkinson M.D., Santoro N. and Urrutia J., "Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters", IEEE Transactions on Communications 34 (1986), No. 6, 614-617.         [ Links ]

[2] Bose R.C., "An affine analogue of Singer's theorem", J. Indian Math. Soc. (N.S.) 6 (1942), 1-15.         [ Links ]

[3] Cilleruelo J., "Sidon sets in ℕ^d", J. Combin. Theory Ser. A 117 (2010), No. 7, 857-871.         [ Links ]

[4] Dimitromanolakis A., "Analysis of the Golomb Ruler and the Sidon set Problems, and Determination of Large, near-optimal Golomb rulers". Thesis (Master), Technical University of Crete, 2002, 118 p.         [ Links ]

[5] Gómez J., "Construcción de conjuntos B_h[g]", Tesis (Maestría), Universidad del Valle, Cali, 2011, 69 p.         [ Links ]

[6] Lindström B., "An inequality for B₂-sequences", J. Combinatorial Theory 6 (1969), 211- 212.         [ Links ]

[7] Martin G. and O'Bryant K., "Constructions of generalized Sidon sets", J. Combin. Theory Ser. A 113 (2006), No. 4, 591-607.         [ Links ]

[8] Ruzsa I., "Solving a linear equation in a set of integers I", Acta Arith. 65 (1993), No. 3, 259-282.         [ Links ]

[9] Singer J., "A theorem infinite projective geometry and some applications to number theory", Trans. Amer. Math. Soc. 43 (1938), No. 3, 377-385.         [ Links ]

[10] Tao T. and Vu V.H., Additive Combinatorics, Cambridge University Press, Cambridge, 2006.         [ Links ]

[11] Trujillo C.A., García G. and Velásquez J.M., "B^± ₂ [g] finite sets", JP J. Algebra Number Theory Appl. 4 (2004), No. 3, 593-604.         [ Links ]

*E-mail: nycaicedob@ut.edu.co.
Received: 31 July 2015, Accepted: 10 November 2015.
To cite this article: Y. Caicedo, C.A. Martos, C.A. Trujillo, g-Golomb, Rev. Integr. Temas Mat. 33 (2015), No. 2, 161-172.