A characterization of finite linear spaces on v points, n2⩽v<(n+1)2, andb=n2+n+3 lines, n⩾10

AbstractCharacterizations of finite linear spaces on v points, n2⩽v<(n+1)2and b=n2+n+1 lines, and on v points, n2⩽v<(n+1)2 and b =n2+n+2 lines have been given by de Witte (unpublished), Stinson (1983) and Batten (1980). Here we extend such a characterization to include the case b=n2+n+3. We show that if n⩾10, then, if it is not a near-pencil, the space is an affine plane of order n less up to three points, with three additional ‘points at infinity’, and three lines, each of size two, on these points