[Paper Review] On Near-MDS Elliptic Codes

Massimo Giulietti|ArXiv.org|Nov 6, 2002

Coding theory and cryptography18 references3 citations

TL;DR

This paper investigates the extendability of near-MDS (NMDS) codes constructed from elliptic curves over finite fields. Using elementary algebraic geometry, it proves that for odd prime powers $ q \geq 121 $, certain $ k $-elliptic codes (for $ k = 3,4,5,6 $) associated with elliptic curves of $ j $-invariant $ \neq 0 $ are non-extendable or not $ h $-extendable for specific $ h $, providing strong non-extendability results that challenge the existence of longer NMDS codes beyond known bounds.

ABSTRACT

The main conjecture on maximum distance separable (MDS) codes states that, execpt for some special cases, the maximum length of a q-ary linear MDS code is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near MDS elliptic code can be extended to a longer near MDS code. Our results are some non-extendability results and an alternative and simpler construction for certain known near MDS elliptic codes.

Motivation & Objective

• To determine whether near-MDS (NMDS) codes derived from elliptic curves can be extended to longer codes.
• To address the open problem of whether NMDS codes can exceed the length $ N_q(1) $, the maximum number of $ \mathbb{F}_q $-rational points on an elliptic curve.
• To provide non-extendability results for $ k $-elliptic codes under specific conditions on $ q $, $ k $, and $ j $-invariant.
• To offer a simplified, elementary construction of known NMDS elliptic codes using basic algebraic geometry.
• To extend the understanding of the structural limitations of NMDS codes beyond the MDS conjecture framework.

Proposed method

• Constructs $ k $-elliptic codes from the rational functions on an elliptic curve $ \mathcal{E} $ over $ \mathbb{F}_q $, using evaluation of functions at $ \mathbb{F}_q $-rational points.
• Employs the Veronese embedding $ \varphi_k $ to map the set of $ \mathbb{F}_q $-rational points of $ \mathcal{E} $ into projective space $ \mathbb{P}^{k-1}(\mathbb{F}_q) $, forming the code's generator matrix.
• Applies geometric arguments involving hyperplanes in $ \mathbb{P}^{k-1} $ to test whether a new point can be added to the code while preserving the NMDS property.
• Uses the order of poles and zeros of rational functions to analyze the linear independence and minimum distance of the code.
• Leverages properties of elliptic curves, such as the number of rational points and the structure of lines intersecting the curve in three rational points, to construct hyperplanes meeting the code in exactly $ k $ points.
• Applies case analysis based on the coordinates of candidate points to show that no hyperplane through a new point $ Q $ can meet the embedded curve in exactly $ k $ rational points, proving non-extendability.

Experimental results

Research questions

• RQ1 Can $ k $-elliptic codes derived from elliptic curves be extended to longer near-MDS codes for $ q \geq 121 $?
• RQ2 What are the structural limitations on extending NMDS codes constructed from elliptic curves over finite fields?
• RQ3 How does the $ j $-invariant of the elliptic curve affect the extendability of its associated $ k $-elliptic codes?
• RQ4 Are there conditions under which $ k $-elliptic codes for $ k = 3,4,5,6 $ are non-extendable or not $ h $-extendable?
• RQ5 Can the non-extendability of such codes be established using elementary algebraic geometry rather than deep geometric machinery?

Key findings

• For $ q \geq 121 $, odd prime power, and $ j(\mathcal{E}) \neq 0 $, the $ k $-elliptic code is non-extendable for $ k = 3 $ and $ k = 6 $, meaning no longer NMDS code can contain it as a subcode.
• For $ k = 4 $, the $ k $-elliptic code is not $ 2 $-extendable, indicating that extending by two coordinates while preserving the NMDS property is impossible.
• For $ k = 5 $, the $ k $-elliptic code is not $ 3 $-extendable, showing that extension by three coordinates fails to preserve the NMDS structure.
• The results are extended to elliptic curves with $ j(\mathcal{E}) = 0 $, provided $ q > 9887 $, $ p > 3 $, and the number of $ \mathbb{F}_q $-rational points is even, with similar non-extendability results holding.
• The method confirms that the construction of longer NMDS codes from elliptic curves is highly constrained, especially for $ k \leq 6 $, and that such codes cannot be extended beyond known bounds under the given conditions.
• The paper provides a simplified, elementary construction of known NMDS elliptic codes, bypassing the need for advanced algebraic geometry used in prior proofs.