Designs, Codes and Cryptography, 4, 57-67 (1994) 9 1994 Kluwer Academic Publishers. Manufactured in The Netherlands. On the [28, 7, 12] Binary Self-Complementary Codes and Their Residuals STEFAN M. DODUNEKOV* Institute of Mathematics, Bulgarian Academy of Sciences, 8 Akad. G. Bonchev Str., 1113 Sofia, Bulgaria SILVIA B. ENCHEVA Department of Mathematics, Technical University of Russe, 7000 Russe, Bulgaria STOYAN N. KAPRALOV Department of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria Communicated by D. Jungnickel Received January 15, 1992; Revised March 25, 1993. Abstract. Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary self- complementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distribution A0 = A28 = 1, A12 = AI6 = 63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified. 1. Introduction An [n, k, d] code is a binary linear code of length n, dimension k and minimum distance d. The code is called self-complementary if it contains the all-one vector. Self-complementary [28, 7, 12] codes were discussed by Brouwer 1981 [3] and Assmus and Key 1989 [1]. Jungnickel and Tonchev 1992 [8] have shown that there exist at least four inequivalent self-complementary [28, 7, 12] codes; they also raised the natural question of the existence of other codes with the same parameters and weight distribution besides those they found. The main result of our paper is that there are exactly four inequivalent binary self- complementary [28, 7, 12] codes. We also classify the codes having the weight distribution of a residual of a self- complementary [28, 7, 12] code. Our approach is based on the residual-code technique. The paper is organized as follows. In Section 2 we present some necessary definitions and preliminary results. The main results are described in Section 3. *This work has been done during the visit of the author as a guest researcher to the Department of Electrical Engineering at Linkrping University, Sweden. The research was partially supported by the Bulgarian National Science Foundation under contracts No. 37/1987 and No. 876/1988.