It is impossible to introduce combinatorial designs and linear codes without basic tools in
certain mathematical areas. In this chapter, we give a brief introduction to cyclotomy, finite …

This article reviews some of the principal and recently-discovered lower and upper bounds
on the maximum size of (n, r)-arcs in PG (2, q), sets of n points with at most r points on a line …

The classical way of extending an [n, k, d] linear code C is to add an overall parity-check
coordinate to each codeword of the linear code C. This extended code, denoted by C‾(− 1) …

The binary Hamming codes with parameters [2 m-1, 2 m-1-m, 3] are perfect. Their extended
codes have parameters [2 m, 2 m-1-m, 4] and are distance-optimal. The first objective of this …

Apart from being an interesting and exciting area in combinatorics with beautiful results,
finite projective spaces or Galois geometries have many applications to coding theory …

Finite Geometries stands out from recent textbooks about the subject of finite geometries by
having a broader scope. The authors thoroughly explain how the subject of finite geometries …

The Erdős‐Rényi and Projective Norm graphs are algebraically defined graphs that have
proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their …

A kind of extended codes of linear codes C with respect to a vector u, denoted by C‾(u), was
introduced by Sun, Ding and Chen (2024)[58]. In this paper, we investigate the weight …

In a previous paper R. Mathon gave a new construction method for maximal arcs in finite
Desarguesian projective planes via closed sets of conics, as well as giving many new …

In this paper we study sets $ X $ of points of both affine and projective spaces over the
Galois field $\mathop {\rm {GF}}(q) $ such that every line of the geometry that is neither …