Few-Weight Linear Sets and Divisibility of Rank-Metric Codes

We study Fq-linear sets whose positive point weights belong to a prescribed finite set A. The integer e=(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.