On sets of vectors of a finite vector space in which every subset of basis size is a basis

Abstract

It is shown that the maximum size of a set S of vectors of a k-dimensional vector space over F q , with the property that every subset of size k is a basis, is at most q + 1 if k ≤ p, and at most q + k − p if q ≥ k ≥ p + 1 ≥ 4, where q = p h and p is prime. Moreover, for k ≤ p, the sets S of maximum size are classified, generalising Beniamino Segre's "arc is a conic" theorem.These results have various implications. One such implication is that a k × (p + 2) matrix, with k ≤ p and entries from F p , has k columns which are linearly dependent. Another is that the uniform matroid of rank r that has a base set of size n ≥ r + 2 is representable over F p if and only if n ≤ p + 1. It also implies that the main conjecture for maximum distance separable codes is true for prime fields; that there are no maximum distance separable linear codes over F p , of dimension at most p, longer than the longest Reed-Solomon codes. The classification implies that the longest maximum distance separable linear codes, whose dimension is bounded above by the characteristic of the field, are Reed-Solomon codes.In the autumn of 2008 while I was visiting Budapest, together with Andras Gács, we formulated the coordinate free version of Segre's lemma of tangents (Lemma 2.1) which is fundamental to this article. I dedicate this work to Andras, whose humour, enthusiasm and brilliance I am grateful to have known.

On sets of vectors of a finite vector space in which every subset of basis size is a basis II

“…Using the Segre product and the lemmas from Section 2 we can give a short proof of [1, Lemma 4.1], the main tool used to prove that |S| ≤ q +1 and classify the case |S| = q +1, for k ≤ p, in [1]. Note that in the second term we can order ∆ ∪ {y} in any way we please without changing the sign since, by Lemma 2.3, interchanging two elements of ∆ ∪ {y} in P D (∆ ∪ {y}, L) changes the sign by (−1) t+1 , exactly the same change occurs when we interchange the same vectors in the product of determinants.…”

“…We shall also prove the conjecture for q prime, which was first proven in [1]. It may help the reader to look at the first four sections of [1], although this article is self-contained (with the exception of the proof of Lemma 2.1) and can be read independently.…”

“…It may help the reader to look at the first four sections of [1], although this article is self-contained (with the exception of the proof of Lemma 2.1) and can be read independently. The proof here is based on the ideas of [1] which themselves are based on the initial idea of Segre in [8].…”

On deep holes of projective Reed-Solomon codes

“…Using recent results of Ball [1] and Ball-De Beule [2] on Conjecture 1.12, we prove Theorem 1.5: Let F q be a finite field of q elements and of odd characteristic p.…”

Pure Asymmetric Quantum MDS Codes From CSS Construction: A Complete Characterization

“…Furthermore, we have the following conjecture which has been shown to be true when q is prime in [2].…”