## Abstract

We study the structure of anticodes in the sum-rank metric for arbitrary fields and matrix blocks of arbitrary sizes. Our main result is a complete classification of optimal linear anticodes. We also compare the cardinality of the ball in the sum-rank metric with that of an optimal linear anticode, showing that the latter is strictly larger over sufficiently large finite fields. Finally, we give examples of parameters for which the largest anticode is neither a ball nor a linear anticode.

Original language | English |
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Pages (from-to) | 80-98 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 643 |

DOIs | |

State | Published - Jun 15 2022 |

### Bibliographical note

Publisher Copyright:© 2022 The Author(s)

### Funding

H. Gluesing-Luerssen was partially supported by the grant #422479 from the Simons Foundation.A. Ravagnani is partially supported by the Dutch Research Council through grant VI.Vidi.203.045.

Funders | Funder number |
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Simons Foundation | |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek | VI.Vidi.203.045 |

## Keywords

- Anticode
- Matrix code
- Sum-rank metric

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics