Abstract The extended coset leader weight enumerator of the generalized Reed–Solomon $$[q+1,q-3,5]_q$$ [ q + 1 , q - 3 , 5 ] q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore, the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse–Weil bound on $${\mathbb F}_q$$ F q -rational points we show that there is a 3-plane containing the passant.

A set $$T\subset {GF(q)}$$ , $$q=p^h$$ is a super-Vandermonde set if $$\sum _{y\in T} y^k=0$$ for $$0< k <|T|$$ . We determine the structure of super-Vandermonde sets of size $$p+1$$ (almost small) and size $$q/p-1$$ (almost large).

It is shown that the parameters of a linear code over ${\mathbb F}_q$ of length $n$, dimension $k$, minimum weight $d$, and maximum weight $m$ satisfy a certain congruence relation. In the case that $q=p$ is a prime, this leads to the bound $m \leq (n-d)p-e(p-1)$, where $e \in \{0,1,\ldots,k-2 \}$ is maximal with the property that ${n-d \choose e} \not\equiv 0 \pmod{p^{k-1-e}}.$ Thus, if $C$ contains a codeword of weight $n$, then $n \geq d/(p-1)+d+e$. The results obtained for linear codes are translated into corresponding results for $(n,t)$-arcs and $t$-fold blocking sets of AG$(k-1,q)$. The bounds obtained in these spaces are better than the known bounds for these geometrical objects for many parameters.

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.

We show that the q-Kneser graph qK 2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q k + q k−1 for k = 3 and for k < q log q − q. We obtain detailed results on maximal cocliques for k = 3.

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown that such graphs, if not complete 3-partite, must have λ ≤ 1.

Improved upper and lower bounds for the size of radar arrays are presented.>