Let [ ] denote a linear code over with length , codimension , and covering radius . We use a modification of constructions of [2 +1 2 3] 2 and [3 +1 3 5] 3 codes ( 5) to produce infinite families of good codes with covering radius 2 and 3 and codimension .

The multicovering radii of a code are recent generalizations of the covering radius of a code. For positive m, the m-covering radius of C is the least radius t such that every m-tuple of vectors is contained in at least one ball of radius t centered at some codeword. In this paper upper bounds are found for the multicovering radii of first order Reed-Muller codes. These bounds generalize the we...

The length function $ \ell_q(r,R) is the smallest of a q $-ary linear code with codimension (redundancy) r and covering radius R $. In this work, new upper bounds on \ell_q(tR+1,R) are obtained in following forms:$ \begin{equation*} \begin{split} &(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; = tR+1,\; t\ge1,\\ &\phantom{(a)\; } q\;{\rm{ \;an\; arbitrary \;prime\; po...

In a recent paper by this author, constructions of linear binary covering codes are considered. In this work, constructions and techniques of the earlier paper are developed and modified for q-ary linear nonbinary covering codes, q 2 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 c...

We consider coverings of a sphere Sn r of radius r with the balls of radius one in an n-dimensional Euclidean space R. Our goal is to minimize the covering density, which defines the average number of the balls covering a point in Sn r . For a growing dimension n, we obtain the covering density at most (n lnn)/2 for any sphere Sn r and the entire space R. This new upper bound reduces two times ...

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric and discrete coupon collector's problem cover times for finite Markov chains, one expects weak concentration bound distribution of time to hold under minimal assumptions. We give two such results, fixed-radius balls other sequentially arriving rand...