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On the generalised rank weights of quasi-cyclic codes

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  • Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual $ 1 $-generator quasi-cyclic codes is almost completely determined by the choice of $ {\mathbb F}_{q^{m}} $.

    Mathematics Subject Classification: Primary: 11T71.


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