Người báo cáo: Đinh Q. Hải (Kent State University)
Thời gian: 9h, Thứ tư 12/7/2017.
Địa điểm: P6, Nhà A14, Viện Toán học, 18 Hoàng Quốc Việt, Hà Nội
Tóm tắt: 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 $rin R$ 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. |
