Partial linear spaces and identifying codes

  • Authors:

  • G. Araujo-Pardo;C. Balbuena;L. Montejano;J. C. Valenzuela

  • Affiliations:

  • Instituto de Matemáticas, Universidad Nacional Autonóma de México, Ciudad Universitaria, México D.F. 04510, Mexico;Departament de Matemítica Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3, E-08034 Barcelona, Spain;Instituto de Matemáticas, Universidad Nacional Autonóma de México, Ciudad Universitaria, México D.F. 04510, Mexico;Departamento de Matemáticas, Universidad de Cádiz, Avda Ramón Puyol s/n, Algeciras (Cádiz), Spain

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2011

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Abstract

Let (P,L,I) be a partial linear space and X@?P@?L. Let us denote (X)"I=@?"x"@?"X{y:yIx} and [X]=(X)"I@?X. With this terminology a partial linear space(P,L,I)is said to admit a(1,@?k)-identifying code if and only if the sets [X] are mutually different for all X@?P@?L with |X|@?k. In this paper we give a characterization of k-regular partial linear spaces admitting a (1,@?k)-identifying code. Equivalently, we give a characterization of k-regular bipartite graphs of girth at least six admitting a (1,@?k)-identifying code. Moreover, we present a family of k-regular partial linear spaces on 2(k-1)^2+k points and 2(k-1)^2+k lines whose incidence graphs do not admit a (1,@?k)-identifying code. Finally, we show that the smallest (k;6)-graphs known up until now for k-1 where k-1 is not a prime power, admit a (1,@?k)-identifying code.