The extended coset leader weight enumerator of a twisted cubic code

Designs, Codes and Cryptography - The extended coset leader weight enumerator of the generalized Reed–Solomon $$[q+1,q-3,5]_q$$ code is computed. In this computation methods in finite...     

Spectral characterization of a graph on the flags of the eleven point biplane

We characterize a 55-point graph by its spectrum ${4^1, (-2)^{10}, (-1 \pm \sqrt{3})^{10}, ((3 \pm \sqrt{5})/2)^{12}}$ . No interlacing is used: examination of tr A ^m for m ≤?7 together with study of the representation in the eigenspace for the eigenvalue ?2 suffices.     

Maximal arcs in Desarguesian planes of odd order do not exist

Forq an odd prime power, and 1<n<q, the Desarguesian planePG(2,q) does not contain an(nq–q+n,n)-arc.Supported by Italian M.U.R.S.T. (Research Group onStrutture geometriche, combinatoria, loro applicazioni) and G.N.S.A.G.A. of C.N.R.     

The Number of Directions Determined by Points in the Three-Dimensional Euclidean Space

   Abstract. Let X be a set of n points in the three-dimensional Euclidean space such that no three points in X are on the same line and there is no plane containing all points in X . An old conjecture states that pairs of points in X determine at least 2n-3 directions. We prove the weaker result that X determines at least 1.75n-2 directions.     

Cocliques in the Kneser graph on the point-hyperplane flags of a projective space

We prove an Erd?s-Ko-Rado-type theorem for the Kneser graph on the point-hyperplane flag in a finite projective space.     

Alternative proof of sine's theorem on the size of a regular polygon in R n with the ℓ∞-metric

It is shown that a regular polygon inR ⁿ with the (2n) ⁿ -metric has at most (2n) ⁿ vertices.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

Locally 4-by-4 grid graphs

We investigate locally grid graphs. The main results are (i) a characterization of the Johnson graphs (and certain quotients of these) as locally grid graphs such that two points at distance 2 have precisely four common neighbors, and (ii) a complete determination of all graphs that are locally a 4 × 4 grid (it turns out that there are four such graphs, with 35, 40, 40, and 70 vertices).