An appropriate version of the linear programming bound of Delsarte for binary codes is used to find explicit upper bounds for A(n, d), with d ~(4.6). These bounds are expected to be at least as good as tAe linear programming bound of Delsarte itself. It is re-established that the Preparata codes are optimal.
1. Introductiun
Let C be a binary code of iength n and minimum distance d. The maximal number of words that C can have is denoted by A(n, d). One of the most basic problems in coding theory is to find good upper bounds for A(n, d). The best results in this respect are obtained by the so-called linear programming (LP-) bound of Dclsarte [3]. For each particular pair (n, d) the LP-bound of A(n, d) is found by solving a related LP-problem. The best known tables containing upper bounds for A(n, d) are based on this method. See e.g. [2]; this paper contains tables for d E {4,6,8,10} and 6 *r_ n < 24.
If ti becomes large, the utility of the LP-method is restricted by the number capacity of the computer which one has at its disposal-That is why it is of interest to look for analytic solutions of the LP-problem of Delsarte. In [l] Best and Brouwer succeeded in solving the LP-problem forf d = 4. Best has also found results for d = 6 (personal communication), but the& have not been published so far. In [6] the authors treated the nonbinary case for d = 4. Let us also mention that in [5] Delsarte's inequalities were used to obtain good asymptotic bounds for the rate of a binary code as a function of its mir:,imum distance.
In this paper we outline a method which gives explicit upper bounds for A(n, dj. These bounds are derived from Delsartc's LP-bound, and it is claimed that our results are at least as good as the (unrestricted) LF-bound. We give explicit formulas for d ~(4,6) only. The ingredients used in this paper are the following. First, Lemma 1, which is equivalent to :,he dual form of the unrestricted . Secondly, some extra inequalities on the distance distribution borrowed from [2]. Thus it appears that the full strength of