We consider two-dimensional $(\lambda_1, \lambda_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes. We present an example to illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code size and code rate.

翻译：我们考虑面积为 $M N$ 的二维 $(\lambda_1, \lambda_2)$-常循环码，其中 $q$ 是素数 $p$ 的某次幂，且满足 $\gcd(M,p)=1$ 与 $\gcd(N,p)=1$。借助公共零点集，我们刻画了二维常循环码。进一步，我们提出了一种算法，利用其本质公共零点集来构造这些码的理想基。我们描述了二维常循环码的对偶码。最后，我们给出了一种用于生成二维常循环码的编码方案。我们通过一个示例说明，在相同码长与码率下，二维常循环码相较于其循环码对应物可能具有更优的最小距离。