Non-additive symmetric rank-metric codes

Abstract

We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd q, we introduce a quotient construction for Sym₃(q) in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on 𝔽_q⁴. This leads to a correspondence with partial ovoids of the elliptic quadric Q^-(5,q). Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in Sym₃(q) of cardinality q⁴+2q³, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd n≥3 and every prime power q>2, we obtain A_q^sym(n,2) ≥ q^(n²−1)/2+q^(n²−3)/2+q^(n²−9)/2, which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension n varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for n=3.

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BibTeX RIS

BibTeX

@misc{messia2026nonadditive,
  title = {{Non-additive symmetric rank-metric codes}},
  author = {Marta Messia and Ferdinando Zullo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18061},
  eprint = {2610.18061},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18061},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Non-additive symmetric rank-metric codes
AU  - Marta Messia
AU  - Ferdinando Zullo
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18061
UR  - https://arxiv.org/abs/2610.18061
AB  - We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd q, we introduce a quotient construction for Sym_3(q) in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on 𝔽_q^4. This leads to a correspondence with partial ovoids of the elliptic quadric Q^-(5,q). Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in Sym_3(q) of cardinality q^4+2q^3, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd n≥3 and every prime power q>2, we obtain A_q^sym(n,2) ≥ q^(n^2−1)/2+q^(n^2−3)/2+q^(n^2−9)/2, which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension n varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for n=3.
ER  -

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