Algebraic structures of repeated-root constacyclic codes of prime power lengths over finite chain rings: applications and generalizations
Speaker: Prof. Đinh Quang Hải (Kent State University, USA)
Time: 9:30 - 11:00, Wednesday June 01, 2022

Venue: Room 612, A6, Institute of Mathematics, VAST

Online: https://meet.google.com/esi-huxm-xqg

Abstract: Since the last 30 years, repeated-root codes have proven to be a very important family of codes. There are many well-known classes of codes whose optimal codes can only be obtained within repeated-root codes. Let $p$ be an arbitrary prime. We investigate 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 $R$, we prove 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 established. 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. Open directions for further generalizations will also be discussed.


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