WEEKLY ACTIVITIES

Time: 9h00, Wednesday, July 12, 2017
Location: Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract: For any given prime $p$, we study the algebraic structure of repeated-root $\lambda$-constacyclic codes of prime power length $p^s$ over a finite commutative chain ring $R$ with maximal ideal $\langle \gamma \rangle$. It is shown that, for any unit $\lambda$ of the chain ring $R$, there always exists an element $r\in R$ such that $\lambda-r^{p^s}$ is not invertible, and furthermore, the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is a local ring with maximal ideal $\langle x-r, \gamma \rangle$.
When there is a unit $\lambda_0$ such that $\lambda=\lambda_0^{p^s}$, the nilpotency index of $x-\lambda_0$ in the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is established. When $\lambda=\lambda_0^{p^s}+\gamma w$, for some unit $w$ of $R$, it is shown that the ambient ring $\frac{R[x]}{\langle x^{p^s}-\lambda \rangle}$ is a chain ring with maximal ideal $\langle x^{p^s}-\lambda_0 \rangle$, which in turn provides structure and sizes of all $\lambda$-constacyclic codes and their duals. Among others, self-dual constacyclic codes are provided. We will also discuss some special cases of the chain ring $R$ that were studied in the literature, as well as some generalizations on the lengths of the codes.
As an application, the Hamming distance $d_H$, homogeneous distance $d_{hm}$, Lee distance $d_L$, Euclidean distance $d_E$, and Rosenbloom-Tsfasman distance $d_{R-T}$, of all negacyclic codes of length $2^s$ over $\mathbb Z_{2^a}$, are completely determined.