🤖 AI Summary
This work investigates the construction of LCD codes with excellent parameters, particularly LCD MDS codes, based on $(\mathcal{L}, \mathcal{P})$-TGRS codes. By carefully selecting evaluation points and imposing specific constraints on the coefficient of $x^{h-1}$ in the twist polynomial, the authors systematically derive two new families of LCD codes from $(\mathcal{L}, \mathcal{P})$-TGRS codes for the first time, subsequently obtaining corresponding LCD MDS codes. The paper also establishes necessary and sufficient conditions under which the constructed codes are AMDS. The proposed approach integrates parity-check matrix analysis, strategic evaluation point selection, and coefficient constraints, with theoretical results validated through concrete examples demonstrating competitive parameters and performance.
📝 Abstract
Twisted generalized Reed-Solomon (TGRS) codes, as a flexible extension of classical generalized Reed-Solomon (GRS) codes, have attracted significant attention in recent years. In this paper, we construct two classes of LCD codes from the $(\mathcal{L},\mathcal{P})$-TGRS code $\mathcal{C}_h$ of length $n$ and dimension $k$, where $\mathcal{L}=\{0,1,\ldots,l\}$ for $l\leq n-k-1$ and $\mathcal{P}=\{h\}$ for $1\leq h\leq k-1$. First, we derive the parity check matrix of $\mathcal{C}_h$ and provide a necessary and sufficient condition for $\mathcal{C}_h$ to be an AMDS code. Then, we construct two classes of LCD codes from $\mathcal{C}_h$ by suitably choosing the evaluation points together with certain restrictions on the coefficient of $x^{h-1}$ in the polynomial associated with the twisting term. From the constructed LCD codes we further obtain two classes of LCD MDS codes. Finally, several examples are presented.