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Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$.
- Source :
-
IEEE Transactions on Information Theory . Jun2020, Vol. 66 Issue 6, p3555-3571. 17p. - Publication Year :
- 2020
-
Abstract
- Let $m$ be an arbitrary positive integer and $D_{8m}$ be the dihedral group of order $8m$ , i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^{2}=1, yxy=x^{-1}\rangle $. Left ideals of the dihedral group algebra $\mathbb {F}_{2}[D_{8m}]$ 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 self-dual binary $[8m,4m]$ -codes such that the dihedral group $D_{8m}$ is 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 $D_{8m}$ -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 $D_{48}$ -codes and left $D_{56}$ -codes respectively. [ABSTRACT FROM AUTHOR]
- Subjects :
- *GROUP algebras
*FINITE rings
*AUTOMORPHISM groups
*AUTOMORPHISMS
*BINARY codes
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 66
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 143457056
- Full Text :
- https://doi.org/10.1109/TIT.2019.2945981