Discrete Mathematics 81 (1990) 103-109 North-Holland 103 zyxwvutsrqponm COMMUNICATION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA PERFECT CODES WITH DISTINCT PROTECTIVE RADII J.M. van den AKKER, J.H. KOOLEN and R.J.M. VAESSENS Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Gommunicated by J.H. van Lint Received 5 December 1989 We consider codes C for which the decoding regions for codewords c are balls B,,(c), where p = r, or p = r,. These are called (r,, r,)-error-correcting codes. If these balls are not only disjoint but also partition the space of all words, then C is called perfect. We are especially interested in codes with the property that centers of balls with the same radius ‘; are at least 2r; + 2 apart (i = 1, 2). These are called bipartite codes. Our main theorem states that a bipartite perfect (r, l)-error-correcting code with r 3 2 must have r = 2 and in fact is obtained from a code with the parameters of a Preparata code. 1. Introduction We shall use standard terminology. A code C of length n over an alphabet Q with q symbols is a subset of Qn. We denote the cardinality JCJ of the code by M; d is the minimum (Hamming-)distance of the code. A ball B,(c) with center c and radius r is defined by B,(c) := {x E Qn ) d(x, c) G r}. (1.1) Suppose that C is the union of two disjoint subcodes Ci and C2 such that the following holds. There are integers r, and rz such that the balls B,(c), where r = ri if c E Cj (i = 1,2) are disjoint. We then call C, with the specified subcodes C, and C2, a (ri, r,)-error-correcting code. Let Mi:= IC,( and let di be the minimum distance of Ci. Then di 2 2ri + 1 (i = 1, 2). If we also define d .=min{d(c,, c2) ( cl E C,, c2 E C,}, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP 1,2. then it is also clear that d I,2 3r*+r,+l. Define r(c) = rj if c E Ci. If c and c’ are codewords with d(c, c’) = r(c) + r(c)) + 1, (1.2) then c and c’ will be called adjucenr. In this way a graph is defined on the vertex set consisting of codewords of C. The code C is called bipartite if the two sets C1 and C2 are independent sets in this graph. 0012-365X/90/$3.50 0 1990, Elsevier Science Publishers B.V. (North-Holland)