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In this talk we analyze one-weight codes in different metrics, focusing on the Hamming, the rank and the sum-rank metric. Bonisoli showed in 1984 that the only one-weight code (up to equivalence and repetition) in the Hamming metric is the simplex code. The classification of one-weight codes in the rank metric is more recent and due to Randrianarisoa in 2019, and also in this case the code is unique (up to equivalence). We will generalize these two  constructions in order to obtain one-weight codes in the sum-rank metric. However, this produces several inequivalent one-weight codes, showing how different the behaviour is in this metric. The main idea to obtain these results is using geometric characterizations for codes in all the three metrics.

This is a joint work with Paolo Santonastaso and Ferdinando Zullo [1].

References

[1] A. Neri, P. Santonastaso, F. Zullo. The geometry of one-weight codes in the sum-rank metric, preprint, arXiv:2112.04989, 2021.