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A projection decoding of a binary extremal self-dual code of length 40

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Abstract

As far as we know, there is no decoding algorithm of any binary self-dual [40, 20, 8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40, 20, 8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two new efficient decoding algorithms for an extremal binary doubly-even self-dual [40, 20, 8] code \(C_{40,1}^{DE}\) by hand with the help of a Hermitian self-dual [10, 5, 4] code \(E_{10}\) over GF(4). The main idea of this decoding is to project codewords of \(C_{40,1}^{DE}\) onto \(E_{10}\) so that it reduces the complexity of the decoding of \(C_{40,1}^{DE}\). The first algorithm is called the representation decoding algorithm. It is based on the pattern of codewords of \(E_{10}\). Using certain automorphisms of \(E_{10}\), we show that only eight types of codewords of \(E_{10}\) can produce all the codewords of \(E_{10}\). The second algorithm is called the syndrome decoding algorithm based on \(E_{10}\). It first solves the syndrome equation in \(E_{10}\) and finds a corresponding binary codeword of \(C_{40,1}^{DE}\).

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Acknowledgments

J.-L. Kim was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005172).

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  1. Department of Mathematics, Sogang University, Seoul, 121-742, South Korea

    Jon-Lark Kim & Nari Lee

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  1. Jon-Lark Kim
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  2. Nari Lee
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Correspondence to Jon-Lark Kim.

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Communicated by C. Mitchell.

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Kim, JL., Lee, N. A projection decoding of a binary extremal self-dual code of length 40. Des. Codes Cryptogr. 83, 589–609 (2017). https://doi.org/10.1007/s10623-016-0253-0

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