Two classes of LCD codes derived from $(\mathcal{L},\mathcal{P})$-TGRS codes

TL;DR

This paper constructs two new classes of LCD codes from twisted generalized Reed-Solomon codes, providing conditions for their optimality and demonstrating their potential as LCD MDS codes through explicit examples.

Contribution

It introduces novel LCD code constructions from $( ext{L}, ext{P})$-TGRS codes, including conditions for AMDS and LCD properties, and extends to LCD MDS codes.

Findings

Derived parity check matrix for $ ext{C}_h$

Established necessary and sufficient conditions for $ ext{C}_h$ to be AMDS

Constructed LCD codes with optimal parameters and provided 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.