Constructions and families of covering codes and saturated sets of points in projective geometry

Abstract

In Davydov (1990), constructions of linear binary covering codes were considered. In the present paper, constructions and techniques of the earlier paper are developed and modified for q-ary linear nonbinary covering codes, q/spl ges/3, and new constructions are proposed. The described constructions design an infinite family of codes with covering radius R based on a starting code of the same covering radius. For arbitrary R/spl ges/2, q/spl ges/3, new infinite families of nonbinary covering codes with good parameters are obtained with the help of an iterative process when constructed codes are the starting codes for the following steps. The table of upper bounds on the length function for codes with q=3, R=2, 3, and codimension up to 24 is given. The author proposes to use saturated sets of points in projective geometries over finite fields as parity check matrices of starting codes. New saturated sets are obtained.