On the Structure of Binary Self-Dual Codes Having an Automorphism of Order a Square of an Odd Prime

New constructions of optimal self-dual binary codes of length 54

“…There is a unique such code with n = 14 and four with n = 16, but for n = 18 it has only been known that the Hermitian self-dual [18,9,8] 4 code is unique [18]. One subproblem of the current study is then to classify the [18,9,6] 4 codes of this type; two different methods were used to obtain and verify this classification result.…”

“…One method used to classify the quaternary Hermitian self-dual [18,9,6] 4 codes is presented in [4]; see also [15]. In general terms, [n, k, d] 4 codes are obtained from 4 codes by adding a new column in all possible ways to the parity check matrices, checking the minimum weight and orthogonality of the new code, and finally removing copies of equivalent codes (an idea is presented in [15] that makes it possible to reject and accept codes without an exhaustive pairwise comparison).…”

“…, 20 have been constructed until now [14]. There are 1,275 inequivalent [54,27,10] self-dual codes with an automorphism of order 9 [6]. We here construct self-dual codes with six new weight enumerators; examples of such codes are given in Tables 3 and 4.…”

“…There is a unique optimal quaternary Hermitian self-dual [18,9,8] 4 code [13]. In this work the quaternary Hermitian self-dual [18,9,6] 4 codes are classified; these are needed in the construction of binary self-dual [54, 27, 10] 2 codes having an automorphism of order 3, which is the topic of this paper.…”

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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.

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