Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN ALTERED GROUP RING CONSTRUCTION OF THE [24, 12, 8] AND [48, 24, 12] TYPE II LINEAR BLOCK CODE

  • Shefali Gupta (Department of Applied Mathematics Delhi Technological University) ;
  • Dinesh Udar (Department of Applied Mathematics Delhi Technological University)
  • Received : 2022.06.02
  • Accepted : 2022.09.05
  • Published : 2023.05.31
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Abstract

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.

Keywords

Acknowledgement

The authors are greatful to the referee for his/her helpful suggestions and comments.

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