Bounds for Covering Codes over Large Alphabets

Abstract

Let K q (n, R) denote the minimum number of codewords in any q-ary code of length n and covering radius R. We collect lower and upper bounds for K q (n, R) where 6 ≤ q ≤ 21 and R ≤ 3. For q ≤ 10, we consider lengths n ≤ 10, and for q ≥ 11, we consider n ≤ 8. This extends earlier results, which have been tabulated for 2 ≤ q ≤ 5. We survey known bounds and obtain some new results as well, also for s-surjective codes, which are closely related to covering codes and utilized in some of the constructions.

On generalized surjective codes

“…We shall prove that it contains a 3-transversal. If P contains the empty set or every column of P contains the same 1-set then the claim follows from σ 4 (4, 2; 0) = 16, see [16]. If P contains a 1-set in one column and a disjoint 1-or 2-set in another column then it is easy to extend them to the desired 3-transversal.…”

Matheuristics for Variants of the Dominating Set Problem

“…These bounds were found proved using theoretical results in Castoldi & Carmelo (2015). In turn, the bounds for the Hamming-covering codes are available in Kéri (2018) and in the book by Cohen et al (1997).…”

Sum-free sets and short covering codes

“…Since the orbit (0, 1, 1) S 3 has three elements and any other orbit Each element in [5] = {1, 2, 3, 4, 5} appears as a coordinate or is a difference of type a − b. Thus we say that P = {(5, 4), (5, 3)} is a δ-covering of [5].…”

“…An overview on covering codes in presented in [2]. Updated tables on these generalized numbers are available in Kéri [5].…”