A note on two-dimensional cyclic and constacyclic codes

Abstract

During the study of the two-dimensional cyclic (TDC) codes of length $n=ls$ over a finite field $\mathbb{F}_q$ where $s=2^k$, Sepasdar and Khashyarmanesh (2016, [11]) arose a problem that the technique used by them to characterize TDC codes of length $n=ls$ does not work for TDC codes of length $3l$. It naturally motivates us to study the TDC codes of other lengths together with $3l$. Further, $(\lambda_1,\lambda_2)$-constacyclic codes are the generalization of constacyclic codes. Thus, we study two-dimensional cyclic codes of length $3l$ and $(\lambda_1,\lambda_2)$-constacyclic codes of length $2l$, respectively over finite fields. Here, the generating set of polynomials for these two-dimensional codes and their duals are obtained. Finally, with the help of our derived results, we have constructed many MDS codes corresponding to the two-dimensional codes.