A group ring construction of the [48,24,12] type II linear block code

A new construction of the self-dual, doubly-even and extremal [48, 24, 12] binary linear block code is given. The construction is much like that of a cyclic code from a polynomial. A zero divisor in a group ring with an underlying dihedral group generates the code. A proof that the code is of minimu...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2012-04, Vol.63 (1), p.29-41
Main Author: McLoughlin, Ian
Format: Article
Language:English
Subjects:
Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
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Summary:A new construction of the self-dual, doubly-even and extremal [48, 24, 12] binary linear block code is given. The construction is much like that of a cyclic code from a polynomial. A zero divisor in a group ring with an underlying dihedral group generates the code. A proof that the code is of minimum distance twelve, without need to resort to computation by computer, is outlined. We also prove the code is self-dual, doubly even and that the code is an ideal in the group ring. The underlying group ring structure is used, which offers a number of useful generator matrices for the code. Interestingly, the construction involves unipotent elements within the group ring, and these lead to the creation of weighing matrices.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-011-9530-0