M. D. S. codes and arcs in projective spaces: a survey

  • Joseph A. Thas


Let C be a code of length k over an alphabet A of size q greather or equal 2. Having chosen m with 2 m  k we impose the following condition on C: no two words agree in as many as m positions. It then follows that |C| qm. If |C|=qm, then C is called a Maximum Distance Separable code (M.D.S. code). A k-arc in PG(n,q) is a set K of k points with k at least n+1 such that no n+1 points lie in a hyperplane. It can be shown that arcs and linear M.D.S. codes are equivalent objects. Here we give a survey of important results on k-arcs, in particular we survey the answers to three fundamental problems on arcs posed by B. Segre in 1955.