• Korean Journal of Mathematics
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• v.10 no.1
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• pp.89-95
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• 2002

We present a simple but new way of decoding the binary Golay code.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.289-317
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• 2002

We associate binary codes to polynomials over fields of characteristic two and show that the binary Golay codes are associated to the Mathieu group polynomials in characteristics two. We give two more polynomials whose Galois group in $M_{12}$ but different self-orthogonal binary codes are associated. Also, we find a family of $M_{24}$-coverings which includes previous ones.

• Journal of the Korean Institute of Telematics and Electronics
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• v.27 no.2
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• pp.9-15
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• 1990

A soft-decision decoding algorithm for linear binary block codes is proposed, for minimizing the block error probability. To compare the proposed algorithm with already established decoding methods, computer simulations are performed for the (7,4)Hamming code and the (23,12) Golay code. The average number of hard-decision decoding is always less then 2, and approaches to 1 when the signal to noise ratio is sufficiently large. These results show that the proposed algorithm reduces the decoding complexity.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.829-844
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• 2023

In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders 2, 3, 4, and 5, and by applying the construction over the binary field and the ring F₂ + uF₂, we obtain extremal binary self-dual codes of various lengths: 12, 16, 20, 24, 32, 40, and 48. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code [24, 12, 8] and the unique Extended Quadratic Residue [48, 24, 12] Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.