## Abstract

We propose the first non-Trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.

Original language | English |
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Pages (from-to) | 5075-5097 |

Number of pages | 23 |

Journal | IEEE Transactions on Information Theory |

Volume | 68 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2022 |

## Keywords

- Decisional sum-rank syndrome decoding problem
- erasure decoding
- generic decoding
- probabilistic hardness reduction
- sum-rank-metric codes