Self-dual binary [8m, 4m]-codes constructed by left ideals of the dihedral group algebra F_2[D_8m]

02/20/2019

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by   Yuan Cao, et al.

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Let m be an arbitrary positive integer and D_8m be a dihedral group of order 8m, i.e., D_8m=〈 x,y| x^4m=1, y^2=1, yxy=x^-1〉. Left ideals of the dihedral group algebra F_2[D_8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D_8m-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left D_8m-codes. These codes make up an important class of binary self-dual codes of length a multiple of 8. Moreover, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing self-dual binary left D_8m-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain 192 extremal self-dual binary [48,24,12]-codes and 728 extremal self-dual binary [56,28,12]-codes.