The maximum number of minimal codewords in long codes

A. Alahmadi, R.E.L. Aldred, R. dela Cruz, P. Solé, Carsten Thomassen

    Research output: Contribution to journalJournal articleResearchpeer-review

    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 2q−p cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than 2p+3log2(3p) edges. We also conclude that an Eulerian (even and connected) graph has at most 2q−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 2q−p+p cycles.
    Original languageEnglish
    JournalDiscrete Applied Mathematics
    Volume161
    Issue number3
    Pages (from-to)424-429
    ISSN0166-218X
    DOIs
    Publication statusPublished - 2013

    Keywords

    • Minimal codewords
    • Intersecting codes
    • Cycle code of graphs

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