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A polymatroid approach to generalized weights of rank metric codes. (English) Zbl 07272714
Summary: We consider the notion of a $$(q, m)$$-polymatroid, due to K. Shiromoto [Des. Codes Cryptography 87, No. 8, 1765–1776 (2019; Zbl 1414.05071)], and the more general notion of $$(q, m)$$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.
##### MSC:
 94B60 Other types of codes 05B35 Combinatorial aspects of matroids and geometric lattices 15A03 Vector spaces, linear dependence, rank, lineability 52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
##### Keywords:
rank metric code; generalized weight; polymatroid; wei duality
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##### References:
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