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.
Riddell and Chan discuss the connection between symmetric and restricted bases @cite_2 . Mossige notes that symmetric bases @math can be efficiently searched by scanning through admissible bases of length @math @cite_0 . For symmetric bases this is sufficient; the second half of a symmetric basis is a mirror image of the first half, and then Rohrbach's theorem ensures that the constructed set @math is a basis for @math . For asymmetric restricted bases, a similar search can be conducted separately for the two halves of the basis (prefix and suffix). However, since Rohrbach's theorem does not apply to asymmetric bases, the construction does not automatically yield a basis for @math . This must be checked separately.
@cite_0 @cite_2
2058902146 1999029321
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