Abstract
Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over \(\mathbb{F}_{2}\), \(\mathbb{F}_{3}\), \(\mathbb{F}_{4}\), \(\mathbb{F}_{5}\), and \(\mathbb{F}_{7}\). In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.
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Acknowledgement
S. Han was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), which is supported by the Ministry of Education, Science and Technology (2010-0007232). J.-L. Kim was partially supported by the Project Completion Grant (year 2011–2012) at the University of Louisville. We also thank Dr. Markus Grassl for informing us the ternary self-dual [76, 38, 18] code in Example 4.3.
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Han, S., Kim, JL. Computational results of duadic double circulant codes. J. Appl. Math. Comput. 40, 33–43 (2012). https://doi.org/10.1007/s12190-012-0543-2
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DOI: https://doi.org/10.1007/s12190-012-0543-2