Roth–Lempel NMDS Codes of Non-Elliptic-Curve Type
Dongchun Han, Cuiling Fan
Abstract
The defect of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k,d]$ </tex-math></inline-formula> linear code is defined as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({\mathcal{ C}})=n-k+1-d$ </tex-math></inline-formula> . Codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({\mathcal{ C}})=0$ </tex-math></inline-formula> are called maximum distance separable (MDS), while codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({\mathcal{ C}})=s({\mathcal{ C}}^{\perp})=1$ </tex-math></inline-formula> are called near maximum distance separable (NMDS). NMDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There have been many constructions of NMDS codes, but most of them are focus on fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , except for constructions from elliptic curves. Roth and Lempel (IEEE Trans. Inf. Theory 1989) constructed a type of linear codes (referred as Roth-Lempel codes), and presented the necessary and sufficient conditions of Roth-Lempel code to be MDS. Especially, they pointed out that the resultant MDS codes is not linearly equivalent to Reed-Solomn codes. In this paper, the NMDS properties of Roth-Lempel codes will be analyzed. We also obtain the necessary and sufficient condition of Roth-Lempel codes to be NMDS, and further completely determine the weight distributions of Roth-Lempel codes with length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q+2$ </tex-math></inline-formula> and dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\leq k\leq q$ </tex-math></inline-formula> . Besides, by analyzing the upper bound for the code lengths of elliptic curve MDS codes, we illustrate the linearly inequivalence of Roth-Lempel NMDS codes and elliptic curve NMDS codes when their corresponding code lengths exceed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4(q+2\sqrt {q}+1)/5+1$ </tex-math></inline-formula> .