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PreprintCC BY 4.0arXiv

Code distances of matrix rank-metric codes under Add-and-Remove transformations

  • Gianira N. Alfarano
  • Rakhi Pratihar
  • Adrien VinΓ§otte
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Abstract

For a matrix rank-metric code π’ž and an integer i, the i-th subcode distance of π’ž is the largest minimum rank distance of an i-dimensional subcode of π’ž. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension β„“s is kept and a space of dimension β„“a is added; this construction is used, with Gabidulin codes, in the MIRANDA signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges β„“s and β„“a. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if β„“s<max{m,n}, then within each block of indices (aβˆ’1)max{m,n}+1,…,amax{m,n} not exceeding dim𝔽q(π’žs), all but the last β„“s subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For MIRANDA, the lower bound on the minimum distance of the dual public code used by the known distinguisher when β„“s=0 is a special case of these results; for the proposed parameters, where β„“s>0, the same bound holds for a subcode of codimension β„“s of the dual public code.

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Cite

BibTeXRIS
BibTeX
@misc{alfarano2026code,
  title = {{Code distances of matrix rank-metric codes under Add-and-Remove transformations}},
  author = {Gianira N. Alfarano and Rakhi Pratihar and Adrien Vinçotte},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03460},
  eprint = {2610.03460},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03460},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Code distances of matrix rank-metric codes under Add-and-Remove transformations
AU  - Gianira N. Alfarano
AU  - Rakhi Pratihar
AU  - Adrien Vinçotte
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03460
UR  - https://arxiv.org/abs/2610.03460
AB  - For a matrix rank-metric code π’ž and an integer i, the i-th subcode distance of π’ž is the largest minimum rank distance of an i-dimensional subcode of π’ž. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension β„“_s is kept and a space of dimension β„“_a is added; this construction is used, with Gabidulin codes, in the MIRANDA signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges β„“_s and β„“_a. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if β„“_s<max{m,n}, then within each block of indices (aβˆ’1)max{m,n}+1,…,amax{m,n} not exceeding dim_𝔽_q(π’ž_s), all but the last β„“_s subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For MIRANDA, the lower bound on the minimum distance of the dual public code used by the known distinguisher when β„“_s=0 is a special case of these results; for the proposed parameters, where β„“_s>0, the same bound holds for a subcode of codimension β„“_s of the dual public code.
ER  -