A Prize Problem in Coding Theory

Kim, Jon-Lark

doi:10.1007/978-3-540-93806-4_23

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…In the following sections we show that this code is the [48, 24, 12] self-dual, doubly-even binary linear block code. 4 In fact, u is only one of a number of elements in the group ring that generates the code in this way. A computer search we conducted using GAP [11] found a total of one hundred and ninety one others of the form 1 + ad where d is a sum of powers of b.…”

Section: A a Imentioning

confidence: 99%

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

McLoughlin

2011

Des. Codes Cryptogr.

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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.

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Section: A a Imentioning

confidence: 99%

“…The problem was posed by Sloane in 1973 [9] and various authors have offered reward for the determination of its existence [4].…”

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confidence: 99%