Two classes of LCD codes derived from $(\mathcal{L},\mathcal{P})$-TGRS codes
Ziwei Zhao, Xiaoni DU, Xingbin Qiao
TL;DR
The paper addresses constructing Euclidean LCD codes from $(\mathcal{L},\mathcal{P})$-TGRS codes and characterizing when the associated $\mathcal{C}_h$ is AMDS. It derives an explicit parity-check matrix $H_h$ for $\mathcal{C}_h$ and proves a precise AMDS criterion involving the twist coefficients $\boldsymbol{\eta}$ and certain determinant conditions. Two LCD constructions are then presented by carefully selecting evaluation points and twisting parameters, yielding LCD codes and, in special cases, LCD MDS codes; one uses a root-of-unity framework with $m(x)=x^n-\lambda$ and a nonzero $r_{h-1}$, the other exploits a symmetry condition $k=(n-l-m)/2$ with a zero-sum constraint. These results broaden the toolkit for LCD and LCD MDS codes based on TGRS codes, providing explicit criteria and illustrating them with examples.
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.