Affine Cartesian codes with complementary duals
Hiram H. López, Felice Manganiello, Gretchen L. Matthews
TL;DR
The paper addresses when generalized affine Cartesian evaluation codes, and their special case generalized Reed-Solomon codes, are LCD, i.e., satisfy $C\cap C^{\perp}=\{0\}$. It develops an LCD-detection framework based on the duality $C_k(\mathcal{A},\mathbf{v})^{\perp}=C_{k'}(\mathcal{A},\mathbf{v}')$ with $k'=\sum_i(n_i-1)-k+1$ and a polynomial $H(\mathbf{X})=\sum_{\mathbf{a}\in\mathcal{A}} v_{\mathbf{a}}^2 L_{\mathbf{a}}(\mathbf{X})$, using an Extended Euclidean Algorithm between $L$ and $H$ to derive concrete degree-based criteria. The authors characterize which generalized Reed-Solomon codes are LCD for arbitrary characteristic and extend the LCD analysis to Cartesian codes on multiple components, showing how LCD-ness of the product relates to LCD status of its factors. They also provide explicit constructions, including LCD MDS codes beyond previous results, and illustrate scalar and point-dependency phenomena through examples. Overall, the work yields practical, algebraically grounded criteria and constructions for LCD codes in a broad class of evaluation codes with applications to cryptography and fault-tolerant systems.
Abstract
A linear code $C$ with the property that $C \cap C^{\perp} = \{0 \}$ is said to be a linear complementary dual, or LCD, code. In this paper, we consider generalized affine Cartesian codes which are LCD. Generalized affine Cartesian codes arise naturally as the duals of affine Cartesian codes in the same way that generalized Reed-Solomon codes arise as duals of Reed-Solomon codes. Generalized affine Cartesian codes are evaluation codes constructed by evaluating multivariate polynomials of bounded degree at points in $m$-dimensional Cartesian set over a finite field $K$ and scaling the coordinates. The LCD property depends on the scalars used. Because Reed-Solomon codes are a special case, we obtain a characterization of those generalized Reed-Solomon codes which are LCD along with the more general result for generalized affine Cartesian codes.