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On sets of vectors of a finite vector space in which every subset of basis size is a basis. (English) Zbl 1241.15002
Summary: It is shown that the maximum size of a set \({ S}\) of vectors of a \(k\)-dimensional vector space over \({\mathbb F}_q\), with the property that every subset of size \(k\) is a basis, is at most \(q+1\), if \(k \leq p\), and at most \(q+k-p\), if \(q \geq k \geq p+1 \geq 4\), where \(q=p^h\) and \(p\) is prime. Moreover, for \(k\leq p\), 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 \(k\times (p+2)\) matrix, with \(k \leq p\) and entries from \({\mathbb F}_p\), has \(k\) columns which are linearly dependent. Another is that the uniform matroid of rank \(r\) that has a base set of size \(n \geq r+2\) is representable over \({\mathbb F}_p\) if and only if \(n \leq p+1\). 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 \({\mathbb F}_p\), 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.

15A03 Vector spaces, linear dependence, rank, lineability
05B35 Combinatorial aspects of matroids and geometric lattices
51E21 Blocking sets, ovals, \(k\)-arcs
94B05 Linear codes, general
Full Text: DOI
[1] Alderson, T., Gács, A.: On the maximality of linear codes. Des. Codes Cryptogr. 53, 59-68 (2009) · Zbl 1172.94642 · doi:10.1007/s10623-009-9293-z
[2] Blokhuis, A., Bruen, A. A., A. Thas, J.: Arcs in P G(n, q), MDS-codes and three fundamental problems of B. Segre-some extensions. Geom. Dedicata 35, 1-11 (1990) · Zbl 0709.51013 · doi:10.1007/BF00147336
[3] Bose, R. C.: Mathematical theory of the symmetrical factorial design. Sankhy\?a 8, 107-166 (1947) · Zbl 0038.09601
[4] Bush, K. A.: Orthogonal arrays of index unity. Ann. Math. Statist. 23, 426-434 (1952) · Zbl 0047.01704 · doi:10.1214/aoms/1177729387
[5] Cherowitzo, W. E.: \alpha -flocks and hyperovals. Geom. Dedicata 72, 221-246 (1998) · Zbl 0930.51007 · doi:10.1023/A:1005022808718
[6] Cherowitzo, W. E., O’Keefe, C. M., Penttila, T.: A unified construction of finite geometries in characteristic two. Adv. Geom. 3, 1-21 (2003) · Zbl 1022.51007 · doi:10.1515/advg.2003.002 · eudml:122947
[7] Cherowitzo, W. E., Penttila, T., Pinneri, I., Royle, G. F.: Flocks and ovals. Geom. Dedicata 60, 17-37 (1996) · Zbl 0855.51008 · doi:10.1007/BF00150865
[8] Glynn, D. G.: Two new sequences of ovals in finite Desarguesian planes of even order. In: Combinatorial Mathematics X, L. R. A. Casse (ed.), Lecture Notes in Math. 1036, Springer, 217-229 (1983) · Zbl 0531.51010
[9] Glynn, D. G.: The non-classical 10-arc of P G(4, 9). Discrete Math. 59, 43-51 (1986) · Zbl 0598.51008 · doi:10.1016/0012-365X(86)90067-1
[10] Hirschfeld, J. W. P.: Rational curves on quadrics over finite fields of characteristic two. Rend. Mat. 3, 773-795 (1971) · Zbl 0239.50012
[11] Hirschfeld, J. W. P., Storme, L.: The packing problem in statistics, coding theory and finite projective spaces: update 2000. In: Finite Geometries, Developments Math. 3, Kluwer, 201- 246 (2001) · Zbl 1025.51012
[12] Hirschfeld, J. W. P., Thas, J. A.: General Galois Geometries. Clarendon Press, Oxford (1991) · Zbl 0789.51001
[13] Kaneta, H., Maruta, T.: An elementary proof and an extension of Thas’ theorem on k-arcs. Math. Proc. Cambridge Philos. Soc. 105, 459-462 (1989) · Zbl 0688.51007 · doi:10.1017/S0305004100077823
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