On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

The study of minimal codewords in linear codes was motivated by Massey who described how minimal codewords of a linear code define access structures for secret sharing schemes. As a consequence of his article, Borissov, Manev, and Nikova initiated the study of minimal codewords in the binary Reed-Muller codes. They counted the number of non-minimal codewords of weight 2d in the binary Reed-Muller codes RM(r, in), and also gave results on the non-minimality of codewords of large weight in the binary Reed-Muller codes RM(r, in). The results of Borissov, Manev, and Nikova regarding the counting of the number of non-minimal codewords of small weight in RM(r,m) were improved by Schillewaert, Storme, and Thas who counted the number of non-minimal codewords of weight smaller than 3d in RM(r,m). This article now presents new results on the non-minimality of large weight codewords in RM(r, m)

Meet & fly : sustainable transport academics and the elephant in the room

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A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results

A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results

Generalised Veroneseans

In \cite{ThasHVM}, a characterization of the finite quadric Veronesean $\mathcal{V}_{n}^{2^{n}}$ by means of properties of the set of its tangent spaces is proved. These tangent spaces form a {\em regular generalised dual arc}. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes