Lp Linear discrepancy of totally unimodular matrices

Let p ∈ [1, ∞[ and cp = maxa ∈ [0, 1]((1 − a)ap + a(1 − a)p)1/p. We prove that the known upper bound lindiscp(A) ⩽ cp for the Lp linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e.,supAlindiscp(A)=cp.We estimate cp=pp+11p+11/p(1+εp) for some εp ∈ [0, 2−p+2], hence cp=1-lnpp(1+o(1)). We also show that an improvement for smaller matrices as in the case of L∞ linear discrepancy cannot be expected. For any p∈Np∈N we give a totally unimodular (p + 1) × p matrix having Lp linear discrepancy greater than pp+11p+11/p.