Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Abstract

Both self-dual codes and maximum distance separable (MDS) codes have nice algebraic structures, theoretical significance, and practical implications. We present two classes of the Hermitian self-dual, MDS, and generalized Reed–Solomon (GRS) codes. Conversely, we prove that the Hermitian self-dual, MDS, and GRS codes must be above two classes for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n =4}$ </tex-math></inline-formula> and conjecture that this result also holds for each even length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {n}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\le {n}\le \text {q+1}$ </tex-math></inline-formula> .