Gauss-Dickson Codes

TL;DR

The paper addresses constructing MDS codes from Jacobi sums associated with Gauss (order $3$) and Dickson (order $5$) diophantine systems over finite fields $\mathbb{F}_q$ with $q=p^{\alpha}$ and primes $p \equiv 1 \pmod{3,5}$. It leverages the arithmetic characterization of Jacobi sums (Katre–Rajwade) to translate diophantine conditions into generator matrices $G=D^t$ that yield 1-error detecting $[2,1,2]$ codes from the Gauss system and 1-error correcting $[4,2,3]$ codes from the Dickson system. A central conjecture (S. A. Katre) about half-row independence and optimal generator matrices is stated and verified for $l=3,5$, with explicit constructions and a decoding example over $\mathbb{F}_{61}$. The work outlines a framework for obtaining higher-order Jacobi-sum–based MDS codes and discusses potential exceptional primes that affect realizability of the codes.

Abstract

Let l be an odd prime. For primes, p \equiv 1 (mod l), Gauss (l = 3) and Dickson (l = 5) considered the Diophantine systems in terms of which cyclotomic numbers of order 3 and 5 were obtained. The aim of this paper is to show how to obtain 1-error detecting [2, 1, 2] code and 1-error correcting [4, 2, 3] code in terms of the solutions of these diophantine systems in the set up of finite fields of q = p^α elements, p \equiv 1 (mod l), l = 3, 5

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 4.1
• Proposition 4.2
• Remark 5.1
• Theorem 5.2
• Example 5.3