## Abstract

A t-(n, d, λ) design over Fq, or a subspace design, is a collection of d-dimensional subspaces of F^{n} _{q} , called blocks, with the property that every t-dimensional subspace of F^{n} _{q} is contained in the same number λ of blocks. A collection of n × m matrices over Fq is said to hold a t-design over Fq if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a t-design over Fq. We show that for F_{q}m-linear vector rank metric codes, the property of a code being maximum rank distance (MRD) is equivalent to its minimal weight codewords holding trivial subspace designs, and show that this characterization does not hold for Fq-linear matrix MRD codes that are not linear over F_{q}m. Finally, using arguments based on covering radius and external distance, we establish various existence results that apply to both the rank and the Hamming metric.

Original language | English |
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Pages (from-to) | 1242-1260 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

Externally published | Yes |

## Keywords

- Assmus-Mattson theorem
- Design over Fq
- MacWilliams identities
- Rank metric code
- Subspace design
- Weight distribution