Optimal linear identifying codes

Identifying codes can be used to locate malfunctioning processors. We say that a code C of length n is a linear (1,≤l)-identifying code if it is a subspace of F₂ⁿ and for all X,Y⊆F₂ⁿ such that |X|, |Y|≤l and X≠Y, we have ∪_x∈X(B(x)∩C)≠∪y∈Y(B(y)∩C). Strongly (1,≤l)-identifying codes are a variant of identifying codes. We determine the cardinalities of optimal linear (1,≤l)-identifying and strongly (1,≤l)-identifying codes in Hamming spaces of any dimension for locating any at most l malfunctioning processors.