Note: The maximum number of minimal codewords in long codes

  • Authors:

  • A. Alahmadi;R. E. L. Aldred;R. Dela Cruz;P. Solé;C. Thomassen

  • Affiliations:

  • Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand;Division of Mathematical Sciences, SPMS, Nanyang Technological University, Singapore and Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines;Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France and Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark and Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provides lower bounds. In this paper, we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightly more than 2^q^-^p cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than 2p+3log"2(3p) edges. We also conclude that an Eulerian (even and connected) graph has at most 2^q^-^p cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as 2^q^-^p+p cycles.