[Paper Review] Construction of MDS Self-dual Codes over Finite Fields

Khawla Labad, Honwei Liu+1 more|arXiv (Cornell University)|Jul 27, 2018

Coding theory and cryptography10 references4 citations

TL;DR

This paper constructs new classes of MDS self-dual codes over finite fields of odd characteristic using generalized Reed-Solomon (GRS) and extended GRS codes. By applying algebraic criteria for self-duality (Corollaries 2.1 and 2.2), the authors derive four theorems that significantly expand the known parameter ranges for such codes, especially for square fields $\mathbb{F}_{q}$ with $q = r^2$, producing up to 713 distinct code lengths compared to 243 in prior work.

ABSTRACT

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For some fixed $q$, our results can produce more classes of MDS self-dual codes than previous works.

Motivation & Objective

• To extend the known existence of MDS self-dual codes over finite fields of odd characteristic.
• To develop systematic constructions using generalized and extended GRS codes that satisfy self-duality conditions.
• To increase the number of parameterized classes of $q$-ary MDS self-dual codes, especially for square fields $\mathbb{F}_{q}$ with $q = r^2$.
• To provide explicit sufficient conditions on parameters ensuring the existence of such codes via algebraic criteria on Lagrange interpolation products.

Proposed method

• Leveraging Corollary 2.1 and Corollary 2.2, which give necessary and sufficient conditions for (extended) GRS codes to be self-dual based on the sign of products $L_{\mathbf{a}}(\alpha_i)$.
• Selecting evaluation points $\mathbf{a}$ and weights $\mathbf{v}$ in $\mathbb{F}_q$ such that $-L_{\mathbf{a}}(\alpha_i) \in \mathbb{F}_q^{*2}$ (the multiplicative group of squares) for all $i$.
• Using structured sets of elements: roots of unity, subspaces of $\mathbb{F}_q$, and cosets of multiplicative subgroups to define $\mathbf{a}$, ensuring symmetry and manageable product expressions.
• Applying properties of quadratic characters and field automorphisms to control the sign of $L_{\mathbf{a}}(\alpha_i)$, particularly when $q = r^2$ or $q = p^m$.
• Deriving explicit bounds on parameters: $t \leq \frac{r-1}{\gcd(r-1,m)}$, $m \mid (q-1)$, and parity conditions on $tm$ or $\frac{q-1}{2t}$ to satisfy self-duality.
• Verifying that the resulting code has dimension $n/2$ and achieves the Singleton bound, confirming MDS and self-dual properties.

Experimental results

Research questions

• RQ1 Can new classes of MDS self-dual codes be constructed over finite fields of odd characteristic using GRS and extended GRS codes?
• RQ2 What parameter conditions on $q$, $n$, and code structure ensure that a GRS or extended GRS code is self-dual?
• RQ3 How do the new constructions compare in terms of the number of distinct code lengths produced compared to previous works?
• RQ4 Can the number of $q$-ary MDS self-dual codes be significantly increased for specific $q = r^2$ by exploiting field structure and multiplicative subgroups?

Key findings

• For $q = r^2$ with $r$ odd, Theorem 1 constructs MDS self-dual codes of length $n = tm$ with $2 \leq t \leq \frac{r-1}{\gcd(r-1,m)}$ and even $\frac{q-1}{m}$, producing more classes than prior works.
• Theorem 2 extends this to $n = tm + 1$ with $tm$ odd, $m \mid (q-1)$, and $t$ in the same range, again yielding new code families.
• Theorem 3 provides constructions for $n = tm + 2$ with $tm$ even and same constraints on $t$ and $m$, further expanding the parameter space.
• Theorem 4 constructs codes of length $n = 2tp^e$ over $\mathbb{F}_{p^m}$ with $2t \mid (p-1)$, $e < m$, and $\frac{q-1}{2t}$ even, enabling new families when $p \equiv 1 \pmod{4}$.
• For $q = 151^2$, the new constructions yield 713 distinct code lengths, compared to 243 in all previous known results listed in Table 1.
• The results generalize and extend prior constructions by [References], particularly for square fields, by introducing new parameter families based on structured evaluation sets and field-theoretic properties.