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

Let p be an odd prime number such that p − 1 = 2 e m 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)/2 i + 1 and λ = (( p − 1)/2 i +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 = 2 m 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.