Existence of rank-metric minimal codes for intermediate lengths

Determine, for given integers q (a prime power), m ≥ 2, k ≥ 2, and lengths n with k + m − 1 ≤ n ≤ 2k + m − 3, whether there exist Fq-linear rank-metric codes C ⊆ Fq^m^n of dimension k that are minimal, namely such that for any nonzero codewords c, c′ ∈ C, inclusion of rank supports σrk(c) ⊆ σrk(c′) holds if and only if c = λc′ for some λ ∈ Fq^m. Establish existence or nonexistence for each of these k − 1 intermediate values of n left unresolved by the known lower bound n ≥ k + m − 1 and the known existence when n ≥ 2k + m − 2.

Background

In the rank metric setting, a code C is an Fq-linear subspace of Fq^m^n with dimension k and rank-support σrk defined by the Fq-row space of the matrix representation of c ∈ C. Minimality means that inclusion of rank supports among nonzero codewords implies Fq^m-scalar proportionality.

Known results give a necessary lower bound n ≥ k + m − 1 for minimality (Corollary 5.10) and guarantee existence of minimal codes for all lengths n ≥ 2k + m − 2 (Theorem 6.11). These bounds leave exactly k − 1 intermediate lengths unresolved. Closing this gap requires determining whether minimal rank-metric codes exist (and possibly providing explicit constructions) for each n in the interval [k + m − 1, 2k + m − 3].