Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$
Yuan Cao, Yonglin Cao, Fang‐Wei Fu, Jian Gao · IEEE Transactions on Information Theory · 2019
Let m be an arbitrary positive integer and D8mbe the dihedral group of order 8m, i.e., D8m= (x, y | x4m= 1, y2= 1, yxy = x-1). Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left D8m-codes. These codes make up an important class of self-dual binary [8m, 4m]-codes such that the dihedral group D8mis necessarily a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left D8m-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary [48, 24, 12]-code and an extremal self-dual binary [56, 28, 12]code from self-dual binary left D48-codes and left D56-codes respectively.