Few-Weight Linear Sets and Divisibility of Rank-Metric Codes
Abstract
We study 𝔽q-linear sets whose positive point weights belong to a prescribed finite set A. The integer e=gcd(A) always divides the rank of the defining subspace, but the divisibility does not by itself imply linearity over 𝔽qe. We show that it does in a broad parameter range. Let LU be an 𝔽q-linear set of rank N spanning PG(h−1,qm), h≥2, without being equal to it, and assume that q≥ m and that the minimum weight of its points is a≥2. Combining the theorem of Csajbók, Marino and Pepe on the maximum field of linearity with the lower bound of Adriaensen and Santonastaso for the size of a linear set containing a point of weight one, we prove that LU is also defined by an 𝔽qt-subspace, for some divisor t of m, whose 𝔽q-dimension is exactly N+t−a. Under a natural gap condition excluding divisors of m in the interval (e,N−e], this forces a=e and the defining subspace to be 𝔽qe-linear. Through geometric duality, the point weights of the dual of a system are the deficiencies m−w of the rank weights w of the associated code. For rank-metric codes in which the subcode generated by the codewords of nonmaximum rank contains codewords of maximum rank, we show that deleting suitable coordinates yields the 𝔽q-expansion of a code over 𝔽qt. We show that the codes in which the codewords of nonmaximum rank generate a subcode without codewords of maximum rank are exactly the direct sums of a simplex code and a code of length smaller than m. We derive field-reduction results for two-weight and three-weight rank-metric codes, and show that no gap condition is needed when m divides the length or when the code has only two nonzero weights. Finally, we study two-dimensional codes of length at most m and provide families of three-weight examples.
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BibTeX
@misc{beule2026fewweight,
title = {{Few-Weight Linear Sets and Divisibility of Rank-Metric Codes}},
author = {Jan De Beule and Leen Demuys and Ferdinando Zullo and Olga Polverino},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08044},
eprint = {2611.08044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08044},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Few-Weight Linear Sets and Divisibility of Rank-Metric Codes
AU - Jan De Beule
AU - Leen Demuys
AU - Ferdinando Zullo
AU - Olga Polverino
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08044
UR - https://arxiv.org/abs/2611.08044
AB - We study 𝔽_q-linear sets whose positive point weights belong to a prescribed finite set A. The integer e=gcd(A) always divides the rank of the defining subspace, but the divisibility does not by itself imply linearity over 𝔽_q^e. We show that it does in a broad parameter range. Let L_U be an 𝔽_q-linear set of rank N spanning PG(h−1,q^m), h≥2, without being equal to it, and assume that q≥ m and that the minimum weight of its points is a≥2. Combining the theorem of Csajbók, Marino and Pepe on the maximum field of linearity with the lower bound of Adriaensen and Santonastaso for the size of a linear set containing a point of weight one, we prove that L_U is also defined by an 𝔽_q^t-subspace, for some divisor t of m, whose 𝔽_q-dimension is exactly N+t−a. Under a natural gap condition excluding divisors of m in the interval (e,N−e], this forces a=e and the defining subspace to be 𝔽_q^e-linear. Through geometric duality, the point weights of the dual of a system are the deficiencies m−w of the rank weights w of the associated code. For rank-metric codes in which the subcode generated by the codewords of nonmaximum rank contains codewords of maximum rank, we show that deleting suitable coordinates yields the 𝔽_q-expansion of a code over 𝔽_q^t. We show that the codes in which the codewords of nonmaximum rank generate a subcode without codewords of maximum rank are exactly the direct sums of a simplex code and a code of length smaller than m. We derive field-reduction results for two-weight and three-weight rank-metric codes, and show that no gap condition is needed when m divides the length or when the code has only two nonzero weights. Finally, we study two-dimensional codes of length at most m and provide families of three-weight examples.
ER -