An additive 2-basis with range n is restricted if its largest element is n 2. Among the restricted 2-bases of given length k, the ones that have the greatest range are extremal restricted. We describe an algorithm that finds the extremal restricted 2-bases of a given length, and we list them for lengths up to k = 41.
New algorithms, based on a very efficient method to compute the h-range, have been used to extend known tables of the extremal h-range, to complete the solution in the case k = 3, and to find a lower bound for the extremal 2-range. By means of a computer search, some extremal additive bases have been constructed which have heretofore been unknown. A set A: 0 = a1 < a2 < * < ak of integers is called a 2-basis for n if each of 0, 1, 2, ... , n can be represented as the sum of two summands from A, with repetition of summands allowed. If the elements of A do not exceed n 2, we call A a restricted 2-basis for n. A more general idea is that of an h-basis, where h summands are used in the representation. However, in this note we shall be dealing only with 2-bases, which we may refer to more briefly as "bases". Rohrbach [5] considered the function n(k) = n2(k), which is the largest integer n for which a basis of k elements exists. An extremal basis is a basis of k elements for n(k). Our object here is to provide a list of extremal bases for 2 < k < 14, most of which have been unknown heretofore. We shall also compare the results of some basis constructions made by Rohrbach. First we give some facts about n(k). By a simple combinatorial argument one obtains