Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.