Projection decoding of some binary optimal linear codes of lengths 36 and 40
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Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes such as cyclic codes, Reed-Solomon codes, and Reed-Muller codes have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, it is a natural question whether which optimal linear codes have an efficient decoding. We show that two binary optimal [36,19,8] linear codes and two binary optimal [40,22,8] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [36,19,8] and [40,22,8] codes. We project them onto the much shorter length linear [9,5,4] and [10, 6, 4] codes over GF(4), respectively. This decoding algorithms, called projection decoding, can correct errors of weight up to 3. These [36,19,8] and [40,22,8] codes respectively have more codewords than any optimal self-dual [36, 18, 8] and [40,20,8] codes for given length and minimum weight, implying that these codes more practical.
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