Construction of Two-Dimensional Cyclic Codes via Cyclotomic Idempotents

This article presents an innovative method for constructing two-dimensional cyclic codes based on the use of primitive idempotents defined via cyclotomic orbits. Our approach exploits the decomposition of the quotient ring \(R = {{{\mathbb{F}}_{q}{\lbrack x,y\rbrack}}/{\langle{x^{s} - 1},{y^{\ell} - 1}\rangle}}\) into a direct product of copies of \({\mathbb{F}}_{q}\) using central primitive idempotents. This decomposition enables the explicit construction of vector space bases and optimized generator matrices for two-dimensional codes. The method incorporates spectral analysis via the discrete Fourier transform, establishing a fundamental link between combinatorial (cyclotomic orbits) and algebraic (primitive idempotents) representations of generator idempotents. We demonstrate that the set \(B = {\{{x^{m}y^{n}e{(x,y)}}\mid{{0 \leq m < k},{0 \leq n < \ell^{\prime}}}\}}\) forms a basis of the two-dimensional cyclic code, with parameters \(\lbrack{s\ell},{k\ell^{\prime}},{{({{s - k} + 1})}{({{\ell - \ell^{\prime}} + 1})}}\rbrack\). The results are validated by explicit examples and generator matrix constructions, offering precise control over code parameters and effectively generalizing BCH-type bounds to the two-dimensional context. This systematic approach fills an important gap in the design of high-performance multidimensional codes.