Light matrices of prime determinant

For A = (a(i, j)) a square integer matrix of prime determinant p, set omega(A) = Sigma(i, j)vertical bar a(i, j)vertical bar. We are interested in the smallest possible value omega(p) for omega(A), and we show that lim(p ->infinity) omega(p)/log(2)(p) = 5/2. We also show that omega(p) <= 2.5 log(2)(p) if and only if p = 2, 7, 13, 37 or a Fermat prime. Our results can also be interpreted as being about addition chains or about presentations of finite cyclic groups.