Shaw, R.; Gordon, N. A. The polynomial degree of the Grassmannian \({\mathcal G}_{1,n,2}\). (English) Zbl 1172.51304 Des. Codes Cryptography 39, No. 2, 289-306 (2006). Summary: For a subset \(\psi\) of \(\text{PG}(N,2)\) a known result states that \(\psi\) has polynomial degree \(\leq r\), \(r \leq N\), if and only if \(\psi\) intersects every \(r\)-flat of \(\text{PG}(N,2)\) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when \(\psi\) is the Grassmannian \({\mathcal G}_{1,n,2} \subset \text{PG}(N,2)\), \(N = \binom{n+1}2 - 1\), to show that for \(n<8\) the polynomial degree of \({\mathcal G}_{1,n,2}\) is \(\binom n2 - 1\). Cited in 3 Documents MSC: 51E20 Combinatorial structures in finite projective spaces 05C90 Applications of graph theory Keywords:polynomial degree Software:DESIGN PDFBibTeX XMLCite \textit{R. Shaw} and \textit{N. A. Gordon}, Des. Codes Cryptography 39, No. 2, 289--306 (2006; Zbl 1172.51304) Full Text: DOI References: [4] David G. Glynn, Johannes G. Maks and L. R. A. (Rey) Casse, The polynomial degree of the Grassmannian \(\mathcal{G} (n,1,q)\) of lines in finite projective space PG(n,q), preprint (July 2003). [7] J. W. P. Hirschfeld and R. Shaw, Projective geometry codes over prime fields, see AMS Contemporary Mathematics Series, Vol. 168, G. Mullen and P. J.-S. Shiue (eds.), Finite Fields: Theory, Applications and Algorithms, Amer. Math. Soc. (1994) pp. 151–163. · Zbl 0872.94053 [9] R. Shaw, Finite geometries and Clifford algebras III, In A. Micali et al. (eds), Proc. of the 2nd Workshop on Clifford Algebras and their Applications in Mathematical Physics, Montpellier, France, (1989); Kluwer Acad. Pubs. (1992) pp. 121–132. [10] R. Shaw, Composition algebras, PG(m,2) and non-split group extensions, In M. A. del Olmo et al. (eds), Proc. of XIXth International Colloquium on Group Theoretical Methods in Physics, Salamanca (1992), Anales de Fisica, Monografias, Vol. 1, CIEMAT / RSEF, Madrid (1993) pp. 467–470. [13] L. H. Soicher, The Design Package for GAP, http://designtheory.org/software/gap_design/. · Zbl 0538.12004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.