A note on the covering ra optimum codes

Bhandari, M.C. and M.S. Garg, A note on the covering radius of optimum codes, Discrete Applied Mathematics 33 (1991) 3-9. This paper gives a lower bound and an upper bound for the covering radius of optimum codes. The upper bound so obtained is better than other known upper bounds, restricted to optimum codes. Optimum codes of covering radius d1 and d2 are shown to be normal. A binary linear code of length n, dimension k and minimum distance d is called an [n, k, d] code. In [IO] Griesmer has shown that for a given k and d, the minimum value of vr (denoted by n(k, d)) for which an [n, k, d] code exists satisfies k-l n(k,d) z c rd/.?.‘l = g(k,d). i=o g(k, d) is called the Griesmer bound; and the code of type [g(k, d), k, d], if it exists, is called a code meeting the Griesmer bound. A code of type [n(k, d), k, d] is called an optimum code. The covering radius R of an [n, k,d] code C is the maximum weight of a coset leader. Many equivalent statements for the covering radius are known [5]. Actual computation of the covering radius, however, is a difficult task; and a number of lower and upper bounds for R have been obtained in [5,9,13]. The best lower bound on the covering radius of an [n, k] code is t[n, k], the minimum covering radius of any [n, k] code [5]. On the other hand the best upper bound known i:; H(n, k, d) = n g(k, d) + d [d/2kl [ 131. It is valid for any [n, k, d] code and implies the Singleton bound [5] and the drd/2kl upper bound for cgde:s meeting the Griesrner bound [4]. The purpose of this paper is to give a lower and an upper bound for the covering radius of optimum codes. The upper bound is shown to be better than the H(n, k,d) bound. A list of codes for which the lower .01166-218X/91/$03.50 (3 1991 Elsevier Science Publishers B.V. All rights reserved 4 M. C. Bhandari, M.S. Garg bound is better than t[n, k] is given in Table 1. The covering radius of a subcode of index 2 is determined and is used to show that optimum codes with covering radius d 1 and d 2 are normal. For definition and results on normal codes we refer the reader to [6]. If C is a [g(k, d), k, d] code and if G is a generator matrix for C, then Busschbach, Gerretzen and van Tilborg have shown the existence of a vector of length k that occurs as a column of G exactly s= rd/gkel 1 times. They also show that no vector appears as a column of G more than s times [4], and that the covering radius R of C satisfies R 5 d[s/21. The following theorem gives a lower bound on R for s = 1. heorem 1. If C is a [g(k, d), k, d] code with s = 1, then R hg(k, d) 2k‘. roof. Let G be a generator matrix for the [g(k, d), k, d] codle C. Since s= 1, columns of G are nonzero and distinct. Hence C is a punctured simplex code & [5] and so R L: R(Sk)-number of columns deleted [S, Corollary I] = 2k-’ l-(2”1 -g(k,d))