Long packing and covering codes

Multicovering bounds from relative covering radii

“…2 Using Corollary 2.2 and the results in Section 12.5 of Cohen, et al [1] we obtain asymptotic results on R m (F n ). For instance, using Theorems 12.5.1 (sphere-covering bound) and 12.5.10.…”

“…, an upper bound on t 1 (n, m) or t 1 [n, k] also gives us a lower bound on R m (C) for any length n code C. Furthermore, by Lemma 1.1, a bound of the form 1 (a, R) = (a, R) ≤ n 0 gives a bound R 2 n−a (C) ≥ n − R for any n ≥ n 0 and any C of length n. Similarly, for every code of length n. Many such bounds are known, and they are well surveyed by Cohen, et al [1]. We summarize the implications for the m-covering radius of F n in several tables.…”

“…For C = F n we obtain the following corollary, which is essentially a restatement of Theorem 19.4.4 of Cohen, et al [1] (cf. also Theorem 19.4.2).…”

“…There are, of course, relationships among these values. These can be proved by straightforward generalizations of the arguments used by Cohen, et al [1]. Lemma 1.1 For positive integers m, n, R, a, k, and n 0 we have 1.…”

“…Then R(C) := R 1 (C) is the covering radius of C. For results on the covering radius, we refer to the book by Cohen, Honkala, Litsyn and Lobstein [1]. For earlier results on multicovering radii, see [4,5,6].…”