On Twisted Generalized Reed-Solomon Codes With <i>ℓ</i> Twists
Haojie Gu, Jun Zhang
Abstract
Abstract-In this paper, we study a class of twisted generalized Reed-Solomon (TGRS) codes with general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell$ </tex-math></inline-formula> twists. A sufficient and necessary condition for the TGRS codes to be MDS or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell$ </tex-math></inline-formula> -MDS <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\ell < \min \{k, n-k\})$ </tex-math></inline-formula> is determined. A sufficient and necessary condition that such a TGRS code is self-dual for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell \leq\left\lfloor\frac{k-1}{3}\right\rfloor$ </tex-math></inline-formula> is also presented. Finally, we give an explicit construction of self-dual TGRS codes.