Consider the symmetric group $S_n$ equipped with the Hamming metric $d_H$. Packing and covering problems in the finite metric space $(S_n,d_H)$ are surveyed, including a combination of both.

The minimal cardinality of a q-ary code of length n and covering radius at most R is denoted by K q (n, R); if we have the additional requirement that the minimum distance be at least d, it is denoted by K q (n, R, d). Obviously, K q (n, R, d) ≥ K q (n, R). In this paper, we study instances for which K q (n, 1, 2) > K q (n, 1) and, in particular, determine K 4 (4, 1, 2) = 28 > 24 = K 4 (4, 1).

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Let K q (n, R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σ q (n, s; r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound K q (n, n − 2) and σ q (n, s; s − 2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on K q (n, n − 2) and to the new bound K q (n, n − 2) 3q − 2n + 2, improving the first iff 5 n < q 2n − 4. We determine K q (q, q − 2) = q − 2 + σ 2 (q, 2; 0) if q 10. Moreover, we obtain the new powerful recursive bound K q+1 (n + 1, R + 1) min{2(q + 1), K q (n, R) + 1}.

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