Designs, Codes and Cryptography

In classical coding theory, different types of extendability results of codes are known. A classical example is the result stating that every $$(4, q^2-1, 3)$$(4,q2-1,3)-code over an alphabet of order q is extendable to a $$(4, q^2, 3)$$(4,q2,3)-code. A ...

Let q be a positive power of a prime number. For arbitrary positive integers h, n, m with n dividing m and arbitrary $$\gamma ,\alpha \in {\mathbb {F}}_{q^m}$$ , Fqm with $$\gamma \ne 0$$ 0 the number of $${\mathbb {F}}_{q^m}$$Fqm-rational points of the ...

A 3-net of order n is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the ...

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the ...

In this article we generalize a theorem of Benson (J Algebra 15:443---454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent ...

Complete $$(k,3)$$(k,3)-arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length $$k$$k and dimension $$3$$3. A class of infinite families of complete $$(k,3)$$(k,3)-arcs in $${\mathrm {...

Let $${\mathbb F}$$F and $${\mathbb F}'$$F be two fields such that $${\mathbb F}'$$F is a quadratic Galois extension of $${\mathbb F}$$F. If $$|{\mathbb F}| \ge 3$$|F| 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar ...

We introduce the notion of t-quasidivisible arc as an (n, w)-arc in $$\hbox {PG}(k-1,q)$$PG(k-1,q) such that every hyperplane has multiplicity congruent to $$n+i$$n+i modulo q, where $$i\in \{0,1,\ldots ,t\}$$i {0,1, ,t}. We prove that every t-...

We consider a code to be a subset of the vertex set of a Hamming graph. The set of $$s$$s-neighbours of a code is the set of vertices, not in the code, at distance $$s$$s from some codeword, but not distance less than $$s$$s from any codeword. A $$2$$2-...

Consider two symmetric $$3 \times 3$$3 3 matrices $$A$$A and $$B$$B with entries in $$GF(q)$$GF(q), for $$q=p^n$$q=pn, $$p$$p an odd prime. The zero sets of $$v^T Av$$vTAv and $$v^T Bv$$vTBv, for $$v \in GF(q)^3$$v GF(q)3 and $$v\ne 0$$v 0, can be ...

Packings of $$\mathrm{PG}(3,q)$$PG(3,q) are closely related to Kirkman's problem of the 15 schoolgirls from 1850 and its generalizations: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange ...

By exploring some geometry of Segre varieties and Veronese varieties, new families of non-linear maximum rank distance codes and optimal constant rank codes are provided.

Given a circle $$C$$C of an inversive plane $${\mathcal {I}}$$I of order $$n$$n, the remaining circles are partitioned into three types according to the number of intersection points with $$C$$C. Let $${\mathcal {S}}$$S be the incidence structure formed ...

We complete the classification of transitive hyperovals with groups of order divisible by $$\textit{four}$$four.

In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective k-subspaces in $$\mathsf {PG}(d,\mathbb {F})$$PG(d,F) are in `higgledy-piggledy arrangement' if they meet each projective subspace of co-...