Abstract
We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field \(\mathbb {F}_{q^m}\), q a prime power, \(m \ge 2\). We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of \([n,n-1]\) constacyclic codes. We characterize polynomial codes of rth rank weight r, and in particular of first rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be.
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Acknowledgements
The early stage of this research by J. Ducoat and F. Oggier was supported by the Singapore National Research Foundation under Research Grant NRF-RF2009-07.
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Communicated by M. Lavrauw.
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Ducoat, J., Oggier, F. Rank weight hierarchy of some classes of polynomial codes. Des. Codes Cryptogr. 91, 1627–1644 (2023). https://doi.org/10.1007/s10623-022-01181-6
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DOI: https://doi.org/10.1007/s10623-022-01181-6