We give an explicit upper bound of the minimal number νT,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in R , n > 2. The asymptotic estimates of νT,n we deduce when n is large are improved further by recent results of Böröczky Jr. and Wintsche on the asymptotic estimates of the minimal number of equal balls of R n covering the sphere S. The optimality of the asympto...

In this paper we construct new family of binary linear completely transitive (and, therefore, completely regular) codes. The covering radius of these codes is growing with the length of the code. In particular, for any integer ρ ≥ 2, there exist two codes in the constructed class of codes with d = 3, covering radius ρ and length ( 4 ρ 2 ) and ( 4 ρ+2 2 ) , respectively. These new completely tra...

In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. The various bounds are then applied to the classes o...

The search for codes of covering radius 1 led Österg̊ard, Quistorff and Wassermann to the OQW method of associating a unique graph to each code [9]. We present results on the structure and existence of OQW-associated graphs. These are used to find an upper bound on the size of a ball of radius 1 around a code of length 3 and minimum distance 2. OQW-associated graphs and non-extendable partial La...

This paper introduces new reduction and torsion codes for an octonary code and determines their basic properties. These could be useful for the classification of self-orthogonal and self dual codes over Z8. We also focus our attention on covering radius problem of octonary codes. In particular, we determine lower and upper bounds of the covering radius of several classes of Repetition codes, Si...

On linear q-ary completely regular codes with ρ = 2 and dual antipodal * Abstract We characterize all linear q-ary completely regular codes with covering radius ρ = 2 when the dual codes are antipodal. These completely regular codes are extensions of 1 linear completely regular codes with covering radius 1, which are all classified. For ρ = 2, we give a list of all such codes known to us. This ...

The covering radius R of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning t[ n, k], the minimal covering radius of any binary code of length n and dimension k. For example r[ n, 41 and t [ n, 51 are determined exactly, and reasonably tight bounds on t[ n, k] are obtained for any k when n is large. These results are found by using sev...

Cohen, G.D., S.N. Litsyn, On the covering radius of Reed-Muller codes, Discrete Mathematics 106/107 (1992) 147-155. We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the ‘essence of Reed-Mul...

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optim...