Des Codes Crypt (2006) 41:101–109 DOI 10.1007/s10623-006-0018-2 New constructions of optimal self-dual binary codes of length 54 Stefka Bouyuklieva · Patric R. J. Östergård Received: 13 December 2005 / Revised: 1 June 2006 / Accepted: 2 June 2006 © Springer Science+Business Media, LLC 2006 Abstract The quaternary Hermitian self-dual [18, 9, 6] 4 codes are classified and used to construct new binary self-dual [54, 27, 10] 2 codes. All self-dual [54, 27, 10] 2 codes obtained have automorphisms of order 3, and six of their weight enumerators have not been previously encountered. Keywords Automorphisms · Cubic codes · Hermitian codes · Self-dual codes AMS Classification 94B05 1 Introduction A linear [n, k] q code C is a k-dimensional subspace of the vector space F n q , where F q is the finite field of q elements. The elements of C are called codewords and the (Ham- ming) weight of a codeword is the number of its nonzero coordinates. The minimum weight d of C is the smallest weight among all nonzero codewords of C; a code C with minimum weight d is called an [n, k, d] q code. A matrix whose rows form a basis of C is called a generator matrix of C. The weight enumerator W(y) of a code C is given by W(y) = ∑ n i=0 A i y i , where A i is the number of codewords of weight i. Let (u, v): F n q × F n q → F q be an inner product in the linear space F n q . The dual code of C is Communicated byV. D. Tonchev. S. Bouyuklieva (B ) Department ofMathematics and Informatics, Veliko Tarnovo University, 5000 Veliko Tarnovo, Bulgaria e-mail: stefka_iliya@yahoo.com P. R. J. Östergård Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000,02015TKK, Helsinki, Finland e-mail: patric.ostergard@tkk.fi