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

Abstract

We study $\mathbb F_q$-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 $\mathbb F_{q^e}$. We show that it does in a broad parameter range. Let $L_U$ be an $\mathbb F_q$-linear set of rank $N$ spanning $PG(h-1,q^m)$, $h\ge2$, without being equal to it, and assume that $q\ge m$ and that the minimum weight of its points is $a\ge2$. 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 $\mathbb F_{q^t}$-subspace, for some divisor $t$ of $m$, whose $\mathbb F_q$-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 $\mathbb F_{q^e}$-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 $\mathbb F_q$-expansion of a code over $\mathbb F_{q^t}$. 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.