IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 1, JANUARY 2012 159 A Putative Doubly Even [72,36,16] Code Does Not Have an Automorphism of Order 9 Nikolay Yankov Abstract—In this paper, we prove that there does not exist a bi- nary self-dual doubly even code with an automorphism of order 9. To do so, we apply a method for constructing binary self-dual codes possessing an automorphism of order for an odd prime . Index Terms—Automorphisms, self-dual codes. I. INTRODUCTION A LINEAR code is a -dimensional subspace of the vector space , where is the finite field of elements. The elements of are called codewords, and the (Hamming) weight of a codeword is the number of the nonzero coor- dinates of . We use to denote the weight of a codeword. The minimum weight of is the smallest weight among all its nonzero codewords, and is called an code. A matrix whose rows form a basis of is called a generator matrix of this code. Every code satisfies the Singleton bound .A code is maximum distance separable or MDS if , and near MDS or NMDS if . For every and from defines the inner product in . The dual code of is . If is called self-orthogonal, and if , we say that is self-dual. We call a binary code self-complementary if it contains the all-ones vector. Every binary self-dual code is self complementary. A self-dual code is doubly even if all codewords have weight divisible by four, and singly even if there is at least one nonzero codeword of weight . Self-dual doubly even codes exist only if is a multiple of eight. The Hermitian inner product on is given by and we denote by the dual of under Hermitian inner product. is Hermitian self-dual if . The weight enumerator of a code is defined as , where is the number of codewords of weight in . Following [10] we say that two linear codes and are permutation equivalent if there is a permutation of coordinates which sends to . The set of coordinate per- mutations that maps a code to itself forms a group denoted Manuscript received April 29, 2011; revised July 02, 2011; accepted July 19, 2011. Date of publication August 22, 2011; date of current version January 06, 2012. This work was supported in part by Shumen University under Project RD-05-157/25.02.2011. The author is with the Faculty of Mathematics and Informatics, Shumen Uni- versity, Shumen 9712, Bulgaria. Communicated by N. Kashyap, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2011.2165829 by . Two codes and of the same length over are equivalent provided there is a monomial matrix and an automorphism of the field such that . The field has an automorphism given by . The set of monomial matrices that maps to itself forms the group called the monomial automorphism group of . The set of maps of the form , where is a mono- mial matrix and is a field automorphism, that map to itself forms the group , called the automorphism group of . In the binary case all three groups are identical. In general, . An automorphism is of type if when decomposed to independent cycles it has cycles of length cycles of length , and fixed points. Obviously, . In [14] Sloane questioned the existence of a doubly even code and found the weight enumerator of such a code: Determining the automorphism group of a code can be fruitful to construct it, so in the papers [2], [3], [5], [8], [11]–[13], [15] the structure and the order of this group was investigated. Recently, O’Brien and Willems refined previous results and showed that the order of the automorphism group of a [72,36,16] doubly even code is 5, 7, 10, 14 or where divides 18 or 24, or the group is [7]. In this paper, we investigate automorphisms of order 9. The proof of the following is completed in Section IV. Main Theorem: There does not exist binary self-dual doubly even codes with an automorphism of order 9. II. CONSTRUCTION METHOD In [6] a method for constructing binary self-dual codes having an automorphism of order , where is an odd prime, was presented. Let be a doubly even self-dual code having an automorphism of order 9. In [4] (Lemma 6) it is proved that is of type , i.e., . Thus, we can assume that (1) Denote by the cycles of length in . Define 0018-9448/$26.00 © 2011 IEEE