The interplay of different metrics for the construction of constant dimension codes

09/15/2021

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by   Sascha Kurz, et al.

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A basic problem for constant dimension codes is to determine the maximum possible size A_q(n,d;k) of a set of k-dimensional subspaces in 𝔽_q^n, called codewords, such that the subspace distance satisfies d_S(U,W):=2k-2(U∩ W)≥ d for all pairs of different codewords U, W. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for A_q(n,d;k) are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases A_q(10,4;5), A_q(11,4;4), A_q(12,6;6), and A_q(15,4;4). We also derive general upper bounds for subcodes arising in those constructions.