On the number of spikes over finite fields

• Authors: Zhaoyang Wu
• Venue: Discrete Mathematics
• Year: 2003

For an integer n ≥ 3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k in {1,2,...,n - 1}, the union of every set of k of these lines has rank k + 1. It is well known that there is a unique binary n-spike for each integer n ≥ 3. In this paper, we first prove that, for each integer n ≥ 3, there are exactly two distinct ternary n-spikes, and there are exactly ⌊(n2 + 6n + 24)/12⌋ quaternary n-spikes. Then we prove that, for each integer n ≥ 4, there are exactly n + 2 + ⌊n/2⌋ quinternary n-spikes and, for each integer n ≥ 18, the number of n-spikes representable over GF(7) is ⌊(2n2 + 6n + 6)/3⌋. Finally, for each q 7, we find the asymptotic value of the number of distinct rank-n spikes that are representable over GF(q).