The maximum number of minimal codewords in long codes

TL;DR

This paper compares theoretical upper bounds and probabilistic lower bounds on the number of minimal codewords in binary codes, and explores similar bounds for cycles in graphs, providing new insights into graph cycle counts.

Contribution

It establishes bounds on the number of cycles in graphs related to minimal codewords, extending previous work and characterizing special graph structures with maximal cycle counts.

Findings

Connected graphs generally have at most 2^{q-p} cycles

Eulerian graphs typically have at most 2^{q-p} cycles

Special graphs (subdivisions of 4-regular graphs) can have up to 2^{q-p}+p cycles

Abstract

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide 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 that $2^{q-p}$ cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than $2p+3\log_2(3p)$ edges. We also conclude that an Eulerian (even) 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.