Galois Hull Dimensions of Gabidulin Codes

2022/11/09 by Islam, Habibul, Horlemann, Anna-Lena · 2 citations
#95B05 #95B15 #95B60 #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2211.05068

Abstract

For a prime power q, an integer m and 0≤ e≤ m-1 we study the e-Galois hull dimension of Gabidulin codes G_k(\boldsymbolα) of length m and dimension k over \mathbbF_q^m. Using a self-dual basis \boldsymbolα of \mathbbF_q^m over \mathbbF_q, we first explicitly compute the hull dimension of G_k(\boldsymbolα). Then a necessary and sufficient condition of G_k(\boldsymbolα) to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of e-Galois (where e=(m)/(2)) self-dual Gabidulin codes of length m for even q, which is in contrast to the known fact that Euclidean self-dual Gabidulin codes do not exist for even q. As an application, we construct two classes of entangled-assisted quantum error-correcting codes (EAQECCs) whose parameters have more flexibility compared to known codes in this context.

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• Properties and Decoding of Twisted GRS Codes and Their Extensions
• Explicit Constructions of Sum-Rank Metric Codes from Quadratic Kummer Extensions

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