Symmetric Designs and Self‐Dual Codes

A construction is given for associating to any symmetric ( v, k, λ) design a self‐dual code of length v + 1 over GF ( p ), where p is any divisor of the square‐free part of k −λ. These codes are then used to obtain a substantial strengthening of a result of Hughes on automorphisms of designs. Specifically, suppose a symmetric ( v, k, λ) design possesses an automorphism σ of odd prime order q . If any prime dividing the square‐free part of k −λ has even (multiplicative) order mod q , then a must fix an odd number of points of the design.