Lower bounds on covering codes via partition matrices

Abstract

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}.

Lower Bounds for $q$-ary Codes with Large Covering Radius

“…Very recently Kéri [5] announced a computer-aided proof of the new bound K 4 (6,3) 10, improving on the bound K 4 (6, 3) 8 due to Chen and Honkala [1]. As an example application of Theorem 2, we give a non-computational proof of this result.…”

“…In a previous paper the first and the third author together with Jörn Quistorff [3] presented a new proof of Rodemich's bound by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound K q (n, n − k) for any k > 2.…”

On generalized surjective codes

“…The generalized surjective codes turned out to be a valuable tool in the theory of covering codes, see [15,17,7].…”

“…Inequality (1) is from [1], (2) is from [6], (3) and (4) and (5) are from [15], (6) and (7) are from [7].…”

“…Furthermore, we only give entries with q < σ q (n, s; r) < q n , confer Section 3. Key to the tables: Unmarked - [5], Theorem 8,11 and 12,[14]; a -(4); b -(5); c - [15]; d - [17]; e -(6); f - [7]; g - (8) j48 − 62 a8 − 24g 9, 6 j96 − 120 a12 − 62g a4 − 16g 9, 7 170 a16 − 62g a7 − 16g 9, 8 256 a32 − 62g a12 − 16g a4m 9, 9 62 16 7 …”