Non-binary linear covering codes

We give a survey of the known works on the topic «nonbinary linear covering codes» and make the analysis of the main results. We consider the linear [[n, n - r]qR covering codes over a field of q elements, q > 2. If covering radius R and codimension (redundancy) r are fixed, the covering problem for codes is to find codes of relatively small length and/or obtaining good upper bounds on the length. The length function ℓq (r, R) is the smallest length of a q-ary linear code with codimension r and covering radius R. In the paper, we give the known lower bound of the length function and formulate two main problems, viz. construction of codes asymptotically achieving the bound and codes having parameters close to the bound. We consider in detail the cases published in literature, where the above problems are solved by constructing infinite families of covering codes. The one-to-one correspondence between linear covering codes and saturating sets in the projective spaces PG(N, q) is mentioned. The qm-concatenating constructions are considered as a tool for constructing the infinite families of covering codes with growing codimension. Some problems of multiple coverings are given.