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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 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 F q -rational points we show that there is a 3-plane containing the passant. [ABSTRACT FROM AUTHOR]

A bipartite biregular (m,n;g) $(m,n;g)$‐graph Γ ${\rm{\Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} $\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$‐regular Moore graphs of even girths g=6,8 $g=6,8$, and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $\left(k,\frac{v-1}{k-1};6\right)$‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$‐bipartite biregular cages for all integers n≥4 $n\ge 4$. Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$‐, (n,n+2;8) $(n,n+2;8)$‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14. [ABSTRACT FROM AUTHOR]

Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth cycles containing e. For a vertex v of Γ let {e1,e2,...,ek} be the set of edges incident to v ordered such that n(e1)≤n(e2)≤⋯ ≤n(ek). Then (n(e1),n(e2),...,n(ek)) is called the signature of v. The graph Γ is said to be girth‐regular if all of its vertices have the same signature. Let Γ be a girth‐regular graph with girth g=2d and signature (a1,a2,...,ak). It is known that in this case we have ak≤(k−1)d. In this paper we show that if ak=(k−1)d−ϵ for some nonnegative integer ϵ

The present Special Issue of Designs, Codes and Cryptography is dedicated to Aart Blokhuis, one of the leading mathematicians in finite geometry and combinatorics. References 1 Ball S, Blokhuis A, Mazzocca F. Maximal arcs in Desarguesian planes of odd order do not exist. The third editor of this special issue, Michel Lavrauw, was a PhD student of Aart between 1997 and 2001. Another challenging problem at the time was to prove the non-existence of maximal arcs in Galois planes of odd order; this was achieved by Aart and his coauthors Simeon Ball and Francesco Mazzocca in [[1]]. [Extracted from the article]

We investigate the upper chromatic number of the hypergraph formed by the points and the k‐dimensional subspaces of PG(n,q); that is, the most number of colors that can be used to color the points so that every k‐subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for t≤38p+1, a small t‐fold (weighted) (n−k)‐blocking set of PG(n,p),p prime, must contain the weighted sum of t not necessarily distinct (n−k)‐spaces. [ABSTRACT FROM AUTHOR]

We show that, for small t, the smallest set that blocks the long secants of the union of t pairwise disjoint Baer subplanes in PG(2,q2) has size t(q+1) and consists of t Baer sublines, and, for large t, the smallest such set has size q2+q+1 and is itself a Baer subplane of PG(2,q2). We also present a stability result in the first case. [ABSTRACT FROM AUTHOR]

A permutation of the point set of the affine space $${{\mathrm{AG}}}(n,q)$$ is called an integral automorphism if it preserves the integral distance defined among the points. In this paper, we complete the classification of the integral automorphisms of $${{\mathrm{AG}}}(n,q)$$ for $$n\ge 3$$ . [ABSTRACT FROM AUTHOR]

A stability theorem says that a nearly extremal object can be obtained from an extremal one by 'small changes'. In this paper, we study the relation of sets having few 0-secants and blocking sets. [ABSTRACT FROM AUTHOR]

A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2, q) be the Desarguesian projective plane over GF( q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF( q), then τ2 PG(2, q))≤ 2( q+( q-1)/( r-1)). For a finite projective plane Π, let [ABSTRACT FROM AUTHOR]

In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given. [ABSTRACT FROM AUTHOR]

The main result of this paper is that point sets of PG( n, q), q = p, p ≥ 7 prime, of size < 3( q + 1)/2 intersecting each line in 1 modulo $${\sqrt[3] q}$$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG( n, p), p ≥ 7 prime, of size < 3( p + 1)/2 with respect to lines are always linear. [ABSTRACT FROM AUTHOR]

The spectrum problem for minimal blocking sets meansthat we wish to determine the possible cardinalities of minimal blockingsets. Besides surveying the results on this problem some new results (ornew proofs) are given. [ABSTRACT FROM AUTHOR]

A subset S of a finite projective plane of order q is called a blocking set if S meets every line but contains no line. For the size of an inclusion-minimal blocking set q+ $$\sqrt q $$ +≤∣ S∣≤ q $$\sqrt q $$ +1 holds ([6]). If q is a square, then in PG(2, q) there are minimal blocking sets with cardinality q $$\sqrt q $$ +1. If q is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3 q points. In the present note we show that in PG(2, q), q≡1 (mod 4) there are minimal blocking sets having more than qlog q/2 points. The blocking sets constructed in this note contain the union of k conics, where k≤log q/2. A slight modification of the construction works for q≡3 (mod 4) and gives the existence of minimal blocking sets of size cqlog q for some constant c. As a by-product we construct minimal blocking sets of cardinality q $$\sqrt q $$ +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of $$\sqrt q $$ parabolas, they are not classical. [ABSTRACT FROM AUTHOR]

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2, q), q even which is not a hyperoval consists of at most q−√ q+1 points. In the first part of our paper we prove this theorem to be sharp for q= s by constructing complete q−√ q+1-arcs. Our construction is based on the cyclic partition of PG(2, q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of ( q−√ q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband's and our constructions. We prove that these constructions result in isomorphic ( q−√ q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF( q ) over GF( q), and on a representation of GF( q) in GF( q ) indicated in Jamison [4]. [ABSTRACT FROM AUTHOR]

The article pays tribute to the late mathematician András Gács who died on October 28, 2009.