L_p Linear discrepancy of totally unimodular matrices

Let p ∈ [1, ∞[ and c_p = max_{a ∈ [0, 1]}((1 - a)a^p + a(1 - a)^p)^1/p. We prove that the known upper bound lindisc_p(A) ≤ c_p for the L_p linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e.,under(sup, A) lindisc_p (A) = c_p .We estimate c_p = frac(p, p + 1) fenced(frac(1, p + 1))^{1 / p} (1 + ε_p) for some ε_p ∈ [0, 2^-p+2], hence c_p = 1 - frac(ln p, p) (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 ∈ N we give a totally unimodular (p + 1) × p matrix having L_p linear discrepancy greater than frac(p, p + 1) fenced(frac(1, p + 1))^{1 / p}.