In this paper, we study the dihedral codes, i.e. the left ideals of $\mathbb{F}_qD_{n}$ in the case $\gcd(q, n) = 1$. An explicit algebraic description of the dihedral codes and their duals is obtained. In addition, a criterion for self-duality of a dihedral code is obtained. Bases, generating and check matrices of dihedral codes are constructed. Given a dihedral code, we consider exterior and interior codes that are induced by cyclic codes. Using this codes, some properties of generating matrices are described and a connection to cyclic code theory is established. In addition, some estimates of code parameters are obtained and several illustrative examples are given.
翻译:在本文中,我们研究了二元编码,即美元=gcd(q,n)=1美元的左翼理想。获得了对二元编码及其双重编码的明确代数描述。此外,还获得了二元编码自我质量的标准。基础、生成和检查二元编码矩阵的构建。根据二元编码,我们考虑了由环流编码引领的外部和内部编码。使用这一编码,描述了生成矩阵的一些特性,并确定了与循环编码理论的联系。此外,还获得了对二元编码参数的一些估计,并列举了几个示例。