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The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an F q -linear rank-metric code over F m×n q of rate R = (1 − ρ)(1 − n m ρ) − ε is shown to be (with high probability) list-decodable up to fractional radius ρ ∈ (0, 1) with lists of size at most Cρ,q ε , where C ρ,q is a constant depending only on ρ and q. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, Håstad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting. IntroductionAt its core, coding theory studies how many elements of a (finite) vector space one can pack subject to the constraint that no two elements are too close. Typically, the notion of closeness is that of Hamming distance, that is, the distance between two vectors is the number of coordinates on which they differ. In a rank-metric code, introduced in [Del78], codewords are matrices over a finite field and the distance between codewords is the rank of their difference. A linear rank-metric code is a subspace of matrices (over the field to which the matrix entries belong) such that every non-zero matrix in the subspace has large rank.Rank-metric codes have found applications in magnetic recording [Rot91], public-key cryptography [GPT91, Loi10, Loi17], and space-time coding [LGB03, LK05]. There has been a resurgence of interest in this topic due to the utility of rank-metric codes and the closely related subspace codes for error-control in random network coding [KK08, SKK08]. Decoding algorithms for rank-metric codes also have connections to the popular topic of low-rank recovery, specifically in a formulation where the task is to recover a matrix H from few inner products H, M with measurement matrices M [FS12]. Finally, the study of rank-metric codes raises additional mathematical and algorithmic

For a vector space F n over a field F, an (η, β)-dimension expander of degree d is a collection of d linear maps Γ j : F n → F n such that for every subspace U of F n of dimension at most ηn, the image of U under all the maps, d j=1 Γ j (U ), has dimension at least β dim(U ). Over a finite field, a random collection of d = O(1) maps Γ j offers excellent "lossless" expansion whp:When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1 + ε with constant degree is a non-trivial goal.We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:Lossless expansion over large fields; more preciselyOptimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ Ω(δd) and η ≥ Ω(1/(δd)) with d = O δ (1), when |F| ≥ n δ . Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Ω(1), 1 + Ω(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with β √ d over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.

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