Secret sharing was introduced by Shamir to distribute a secret among a group of participants. Later, Massey generalized this approach using minimal linear codes. Recent research has extensively explored the basic properties, code parameters, and constructions of minimal linear codes from algebraic, combinatorial, and geometric perspectives. In this paper, we further generalize Massey’s scheme to multi-secret sharing by introducinghigher-order minimal linear codes. We present some fundamental properties of these codes, including two necessary and sufficient conditions forrth-order minimality. These conditions generalize the Heng–Ding–Zhou and the Lu–Wu–Cao conditions for ordinary minimal linear codes, respectively. We also derive bounds on the parameters of higher-order minimal linear codes. Importantly, we prove that these codes are asymptotically good, exhibiting both non-vanishing rate and relative distance as the code length tends to infinity. A geometric characterization establishes an equivalence betweenrth-order minimal linear codes and cuttingr-blocking sets in projective geometry. Finally, we provide higher-order minimal linear codes using punctured simplex codes, Kronecker products and subfield subcodes, respectively.